# Octant projection

The octant projection[1][2][3] or octants projection, is a type of map projection[4] proposed the first time, in 1508, by Leonardo da Vinci in his Codex Atlanticus.[5] Leonardo's authorship would be demonstrated by Christopher Tyler, who stated "For those projections dated later than 1508,[6] his drawings should be effectively considered the original precursors."[5][7]

The same page of the Codex contains sketches of eight other projections of the globe (those known in the late fifteenth century) studied by Leonardo, including Ptolemy's conical planisphere projection and Roselli's pseudocylindric projection. [7]

## Description

The octant projection is the first known polyhedral map projection. It is neither conformal nor equal-area.[8] In it, the spherical surface of the earth is divided into eight octants, each flattened into the shape of a Reuleaux triangle bound by circular arcs. If transferred to an elastic support, it would be possible to cover with them the surface of a model of the earth's globe.[9][10]

Each of the eight triangles has as one of its sides one fourth of the equator. The remaining two sides are part of the two meridians that with the equator dissect the globe in the eight octants.[11]

## History of authorship research

Although Leonardo's first description of the octant projection has been proved by Tyler,[5] who decided to treat separately Leonardo's projection authorship (1508) from Leonardo's map authorship (1514), the other authors before him treat together the authorship of both map and projection, for they speak about "the eighth of a supposed globe represented in a plane" or about "globe sections" (Harrisse) or others about "gores", which are in fact a projection of the globe.

So, bearing in mind the fact that Tyler was the first scholar to mention the sketch of this projection in Codex Atlanticus in 2017, the authorship of the map it is not universally accepted, with some authors being completely against any minimal contribution from Leonardo, such as Henry Harrisse (1892),[13] or Eugène Müntz (1899 – citing Harrisse authority from 1892, although none of them talks about the projection's sketch in Codex Atlanticus).[14]

Other scholars accept explicitly both (map and projection: "the eight of a supposed globe represented in a plane"), completely as a Leonardo's work, describing the projection as the first of this type, among them, R. H. Major (1865) in his work Memoir on a mappemonde by Leonardo da Vinci, being the earliest map hitherto known containing the name of America,[15] Grothe,[16] the Enciclopedia universal ilustrada europeo-americana (1934),[9] Snyder in his book Flattening the Earth (1993),[10] Christoher Tyler in his paper (2014) "Leonardo da Vinci’s World Map",[5] José Luis Espejo in his book (2012) Los mensajes ocultos de Leonardo Da Vinci,[17] or David Bower in his work (2012) "The unusual projection for one of John Dee's maps of 1580".[12]

Others also accept explicitly both (map and projection) as authentic, although leaving in the air Leonardo's direct hand, giving the authorship of the work to one of his disciples as Nordenskjold states in his book Facsimile-Atlas (1889) confirmed by Dutton (1995) and many others: "on account of the remarkable projection..not by Leonardo himself, but by some ignorant clerk",[18] or Keunig (1955) being more precise: "by one of his followers at his direction".[19]

## References

1. ^ Art of geography (archive.org) 2016-03-04
2. ^ Flamingpear map-projections
3. ^ Csiss.org-map-projections
4. ^ Geological Survey (U.S.) (1988). Bibliography of map projections. U.S. G.P.O. pp. 73–.
5. ^ a b c d Tyler, C.W. (2017). "Leonardo da Vinci's World Map" (PDF). Journal of the International Map Collector's Society (149 Summer): 21–31.
6. ^ Mapa de Juan de la Cosa (pág.27)
7. ^ a b Tyler, Christofer (2014). Leonardo da Vinci's World Map (2014) (pdf). London: J.B. Nicholls and Sons. Retrieved 20 September 2015.
8. ^ [A conformal projection has the following properties: the scale at any point is the same in all directions.. and the angle between any two lines on the map must be the same as on the earth. An equal area projection preserves relationships between corresponding areas..]
9. ^ a b
10. ^ a b Snyder, John P. (1997), Flattening the Earth: Two Thousand Years of Map Projections, University of Chicago Press, p. 40, ISBN 978-0-226-76747-5.
11. ^ John P. Snyder (5 December 1997). Flattening the Earth: Two Thousand Years of Map Projections. University of Chicago Press. pp. 40–. ISBN 978-0-226-76747-5.
12. ^ a b Bower, David I. (February 2012), "The unusual projection for one of John Dee's maps of 1580" (PDF), The Cartographic Journal, 49 (1): 55–61, doi:10.1179/1743277411y.0000000015, S2CID 129873912.
13. ^ Henry Harrisse (1872). Bibliotheca Americana Vetustissima: A Description of Works Relating to America Published Between the Years 1492 and 1551. Librairie Tross.
14. ^ Eugène Müntz (8 May 2012). Leonardo da Vinci. Parkstone International. pp. 97–. ISBN 978-1-78160-387-1.
15. ^ Major, Richard Henry (1865). Memoir on a mappemonde by Leonardo da Vinci, being the earliest map hitherto known containing the name of America, now in the Royal Collection at Windsor (pdf). London: J.B. Nicholls and Sons. Retrieved 21 March 2015.
16. ^ Grothe, Hermann (1874). Leonardo da Vinci als Ingenieur and Philosoph (pdf). Berlin: Berlin: Nicolaische. Retrieved 20 September 2015.
17. ^ José Luis Espejo Pérez (2012). Los mensajes ocultos de Leonardo Da Vinci. Editorial Base. pp. 60–67. ISBN 978-84-15706-00-7.
18. ^ Geoffrey H. Dutton (1999). A Hierarchical Coordinate System for Geoprocessing and Cartography. Springer. ISBN 978-3-540-64980-9.
19. ^ Keuning, Johannes (January 1955), "The history of geographical map projections until 1600", Imago Mundi, 12 (1): 1–24, doi:10.1080/03085695508592085, JSTOR 1150090.