octant projection

File:Cahill butterfly projection3.jpg

The octant projection is the first known polyhedral map projection. It is neither conformal nor equal-area.[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..] 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.{{cite book|title=Enciclopedia universal ilustrada europeo-americana:"..así el mapa de leonardo en ocho segmentos estaba destinado a un globo.."|url=https://books.google.com/books?id=qgQjAQAAMAAJ&q=as%C3%AD+el+mapa+de+leonardo+en+ocho+segmentos+estaba+destinado+a+un+globo|year=1934|publisher=J. Espasa}}{{citation|title=Flattening the Earth: Two Thousand Years of Map Projections|first=John P.|last=Snyder|publisher=University of Chicago Press|year=1997|isbn=978-0-226-76747-5|page=40|url=https://books.google.com/books?id=0UzjTJ4w9yEC&pg=PA40}}.

The eight triangles are oriented in a similar way as per two four-leaf clovers{{cite web |date=2024-06-07 |title=Da Vinci's octant projection |url=https://observablehq.com/@fil/da-vincis-octant-projection |access-date=2024-08-20 |website=Observable}} side by side (fleurons),{{cite web |date=2018-11-01 |title=La projection en octants de Léonard de Vinci – Visionscarto |url=https://www.visionscarto.net/projection-octants-da-vinci |access-date=2024-08-20 |publisher=Visionscarto |language=fr}} being the earth poles in the center of each clove. One of the sides of the eight triangles, (the one opposite to the center of the pseudo clover), is one fourth of the equator, the remaining two (those that converge to the center of the pseudo clover), are part of the two meridians that with the equator dissect the globe in the eight octants.{{cite book |author=John P. Snyder |url=https://books.google.com/books?id=0UzjTJ4w9yEC&dq=world+in+eight+spherical+octants&pg=PA40 |title=Flattening the Earth: Two Thousand Years of Map Projections |date=5 December 1997 |publisher=University of Chicago Press |isbn=978-0-226-76747-5 |pages=40–}}

File:DaVinci Codex-Atlanticus-Projections.JPG with sketches of eight other projections of the globe being studied by Leonardo]]

Projections also based on the Reuleaux triangle were published by:

• 1549 – Oronce Finé
• 1556 – Le Testu
• 1580 – John Dee
• 1616 – Nicolaas Geelkercken
• 1894 – Fiorini
• 1909 – Bernard J. S. Cahill
• 1916 – Anthiaume
• 1938 – Uhden
• 1955 – Keuning
• 1975 – Cahill–Keyes

Although Leonardo's first description of the octant projection has been proved by Tyler, 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),{{cite book|author=Henry Harrisse|title=Bibliotheca Americana Vetustissima: A Description of Works Relating to America Published Between the Years 1492 and 1551|url=https://archive.org/details/cihm_05617|year=1872|publisher=Librairie Tross}} or Eugène Müntz (1899 – citing Harrisse authority from 1892, although none of them talks about the projection's sketch in Codex Atlanticus).{{cite book|author=Eugène Müntz|title=Leonardo da Vinci|url=https://books.google.com/books?id=cbPbWHDtU1sC&dq=Ximenes+had+developed+a+letter+of+Leonardo+to+Christopher+Columbus+from+1473&pg=PA97|date=8 May 2012|publisher=Parkstone International|isbn=978-1-78160-387-1|pages=97–}}

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,{{cite book|last1=Major|first1=Richard Henry|title=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.|date=1865|publisher=J.B. Nicholls and Sons|location=London|url=https://archive.org/details/cu31924029955550|access-date=21 March 2015|format=pdf}} Grothe,{{cite book|last1=Grothe|first1=Hermann|title=Leonardo da Vinci als Ingenieur and Philosoph|date=1874|publisher=Berlin: Nicolaische|location=Berlin |url=https://archive.org/details/leonardodavincia00grot|access-date=20 September 2015|format=pdf}} the Enciclopedia universal ilustrada europeo-americana (1934), Snyder in his book Flattening the Earth (1993), Christopher Tyler in his paper (2014) "Leonardo da Vinci's World Map", José Luis Espejo in his book (2012) Los mensajes ocultos de Leonardo Da Vinci,{{cite book|author=José Luis Espejo Pérez|title=Los mensajes ocultos de Leonardo Da Vinci|url=https://books.google.com/books?id=q6aqNAEACAAJ|pages=60–67|year=2012|publisher=Editorial Base|isbn=978-84-15706-00-7}} or David Bower in his work (2012) "The unusual projection for one of John Dee's maps of 1580".{{citation

| last = Bower | first = David I.

| date = February 2012

| doi = 10.1179/1743277411y.0000000015

| issue = 1

| journal = The Cartographic Journal

| pages = 55–61

| title = The unusual projection for one of John Dee's maps of 1580

| url = http://dibower.co.uk/data/documents/Dee-projn-paper.pdf

| volume = 49| s2cid = 129873912

}}.

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",{{cite book|author=Geoffrey H. Dutton|title=A Hierarchical Coordinate System for Geoprocessing and Cartography|url=https://books.google.com/books?id=5VLuAAAAMAAJ&q=nordenskjold+It+is,+however,+worthy+of+attention+from+a+cartographical+point+of+view,+not+merely+on+account+of+the+remarkable+projection|year=1999|publisher=Springer|isbn=978-3-540-64980-9}} or Keunig (1955) being more precise: "by one of his followers at his direction".{{citation

| last = Keuning | first = Johannes

| date = January 1955

| doi = 10.1080/03085695508592085

| issue = 1

| journal = Imago Mundi

| jstor = 1150090

| pages = 1–24

| title = The history of geographical map projections until 1600

| volume = 12}}.

File:Leonardo da Vinci’s Mappamundi.jpg's octant projection in eight octants with Reuleaux triangle's shape]]

File:Nicolaas-Geelkercken Projection 1616.JPG's shape]]

File:Cahill butterfly projection-1915.jpg

• World map
• List of map projections
• List of works by Leonardo da Vinci
• Leonardo's world map
• Rhumbline network
• Waterman butterfly projection
• Waterman polyhedron
• Bernard J. S. Cahill

{{reflist|2}}

{{Commons category|Reuleaux triangles}}

• [https://c3jud0.wordpress.com/2008/08/24/proyecciones-cartograficas/ Proyecciones-cartograficas]

{{Map projections}}

{{Leonardo da Vinci}}

Category:Map projections

Category:Works attributed to Leonardo da Vinci