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Four is a figurate number as it can be represented by four dots arranged in a square. The next highest square is nine. Thus, the gnomon (sense 5) between four and nine is five(represented by the blue dots).
Borrowed from Frenchgnomon, or directly from its etymonLatingnōmōn, or directly from its etymon Ancient Greekγνώμων(gnṓmōn, “discerner, interpreter; carpenter’s square; gnomon of a sundial; (geometry) gnomon”), from γιγνώσκω(gignṓskō, “to be aware of; to perceive; to know”), ultimately from Proto-Indo-European*ǵneh₃-(“to know”); the word is thus related to know.
The geometry sense (sense 4) is from the resemblance of the plane figure to a carpenter’s square. Similarly, a gnomon in mathematics (sense 5) is also shaped like a carpenter’s square when depictedpictorially if the figurate numbers are squares.
1700, Samuel Foster, “ The Ninth Tractate. Appendix.] Chap. IV. A General and Most Easie Way to Project Hour-lines upon All Kinds of Superfices without any Regard had to Their Standing, either in respect of Declination or Inclination”, in William Leybourn, compiler, Dialling, Plain, Concave, Convex, Projective, Reflective, Refractive. Shewing, how to Make All such Dials, and to Adorn Them with All Useful Furniture Relating to the Course of the Sun;, 2nd edition, London: Printed by J. Matthews, for Awnsham and John Churchill, →OCLC, page 83:
Let a Gnomon, being firſt ſharpened into a Point, be ſhaped, and faſtned in ſuch wiſe, that it no way hinder either the draught of the Horizontal Line, or the Point of the ſhadow from having free access to the Dial at all times of the Year. […] Upon the Superfices of the Dial, obſerve the Point of the Shadow of the Gnomon (making a mark at it) and the Sun's Altitude, both of them at the ſame inſtant of time.
1758 March, “An Account of an Ancient Astronomical Instrument, of Immense Dimensions, lately Restored to Use”, in Sylvanus Urban [pseudonym; Edward Cave], editor, The Gentleman’s Magazine, and Historical Chronicle, volume XXVIII, London: Printed by D Henry and R. Cave,, published January 1755, →OCLC, page 99, column 1:
The moſt celebrated gnomons, for examining the courſe of the ſun, which are exiſting at this time, have been ſuppoſed to be that in the church of St Petronio at Bologna, 89 Engliſh feet high; that of the Chartreux at Rome, about 67 feet; and that of St Sulpice at Paris, 86 feet. […] [T]he gnomon of the cathedral of that city [Florence], erected by [Paolo dal Pozzo] Toscanelli, in the courſe of the 16th century, whoſe height exceeds that of the three before-mentioned, taken all together, being near 300 Engliſh feet, and higher than our London monument by almoſt a third part.
Approximate scales can be placed on near-field objects in pictures containing the gnomon by using an ellipse template. The Apollo 16gnomon[…] stands on three legs that, on a hard surface, define a circle 61.7 cm in diameter.
1659, Joseph Moxon, “Prob XVIII. Another Way to Find the Height of the Pole by the Globe; if the Place of the Sun be Given; and also to Find the Hour of the Day, and Azimuth, and Almicantar of the Sun.”, in A Tutor to Astronomie and Geographie: Or An Easie and Speedy Way to Know the Use of Both the Globes, Cœlestial and Terrestrial. In Six Books., London: Printed by Joseph Moxon; , →OCLC, 2nd book (Shewing the Practical Use of the Globes. Applying Them to the Solution of Astronomical and Geographical Problemes.), page 56:
This muſt be performed by help of a Spherick Gnomon, (as Blaew [Willem Janszoon Blaeu] calls it,) which is a ſmall Pin or Needle fixed perpendicularly into a ſmal Baſis with an hollow concave bottom, that it may ſtand upon the convexity of the Globe.
1791, William Robertson, “Section II. Intercourse with India, from the Establishment of the Roman Dominion in Egypt, to the Conquest of that Kingdom by the Mahomedans”, in An Historical Disquisition Concerning the Knowledge which the Ancients had of India;, London: Printed for A Strahan, and T Cadell; Edinburgh: E. Balfour, →OCLC, pages 78–79:
In order to aſcertain the latitude of any place, the ancients obſerved the meridian altitude of the ſun, either by means of the ſhadow of a perpendicular gnomon, or by means of an aſtrolabe, from which it was eaſy to compute how many degrees and minutes the place of obſervation was diſtant from the Equator.
Indexof a globe, the little style or gnomon, which being fixed on the pole of the globe, and turning round with it, points out the hours upon the hour circle.
If a Parallelogram be divided into four leſſer ones, by two Lines interſecting each other; and one of theſe Parallelograms be retrench'd, or taken away; the other three will make a Gnomon, ordinarily call'd a Square.
[page 16] The figurate number's gnomon is therefore in the general case. A trivial family of numbers, those for which , might be called linear. […] [page 17] Clearly, a figurate number of rank is equal to the sum of its first gnomons, and in particular the sum of the firstodd numbers is equal to.