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Cahill Butterfly Map 1909

Cahill-Keyes M-layout world map silhouette including
Cahill 1909
Cahill-Keyes 1975

Gene Keyes
B.J.S. Cahill's World Map Variants:
1) Conformal

In the 30 years after Cahill published his initial Butterfly map in 1909, he also produced several "Variants" as well as other false starts, with a confusing array of nomenclature and neologisms. However, besides the 1909 original, Cahill settled on three main types, the clearest illustration of which is in this [1934] offprint from Nelson's Encyclopedia:
1) Conformal
2) Equal Area
3) Gnomonic
Cahill 3-in-1
Source: B.J.S. Cahill, "Projection"(Nelson's Encyclopedia, [1934], offprint;
 also in Cahill, "The Butterfly Map: Octahedral System of Projection" (Pacific Marine Review, 1938-09) p. 42-43.
Scanned and enlarged by GK from a duplicate offprint received from
B.J.S. Cahill Collection at The Bancroft Library, University of California, Berkeley.

Note: on an earlier page, I showed a smaller enlargement of this image at the same scale of
1/200 M as a set of other Butterfly maps. Here, I have made it yet larger for emphasis of the Three Variants.

A conformal map is one in which meridians and parallels cross one another at right angles — as on a globe, but much more problematic when it comes to devising a flat world map.

On Cahill's earliest map, the graticule was a very well-drafted custom-made assembly — he was an architect, remember —, but it was derided by ultra-mathematical cartographers. So Cahill set about making some more rigorously mathematical variants, and even hired the prominent mathematician Oscar S. Adams to compute their graticular distributions at five-degree intervals for all three main variants. Meanwhile, Cahill had not forgotten what I consider his foremost principle:
The creation of a master plan of the world is a matter of design first and foremost, the design of a frame within which intensive mathematical details shall be afterwards subordinated. (1928)
In most of his examples, Cahilll's Conformal is embedded in equilateral triangles, unlike the truncated ones of Variant 2, the Equal Area (and the curvilinear octants of his 1909 original).  Below is an undated [mid 1920s?] Cahill Conformal draft in the North Pacific Butterfly orientation, one of a number of illustrations ["Fig. 5"] for an unpublished article or book manuscript. Like his earlier maps, it has his dubious 7.5 x 10º graticule, which conceals the split geocells along the perimeter meridian. (By 1927, Cahill was at last showing a 5º graticule.)

Source: B.J.S. Cahill Collection at The Bancroft Library,
 University of California, Berkeley;
Xeroxed, then scanned and reduced to 98% of original for a scale of 1/200 M
by GK.
Cahill Conformal N. Pacific

The next one [mid 1920s?] is done at 5º resolution, betraying the sore-thumb two-and-a-half-degree geocells along the outer perimeter. (In this version, however, Cahill has omitted the internal northern triangle sides, concealing the half-degree problem.in those areas.)

Source: B.J.S. Cahill Collection at The Bancroft Library,
University of California, Berkeley;
Xeroxed, then scanned and reduced to 75% of original for a scale of 1/200 M
by GK. (Larger copies further below.)
Cahill Conformal 5 degrees

This 1924 drawing is perhaps the only instance in which Cahill showed a Conformal graticule inside curvilinear octants, whereas his other Conformals were all done with equilateral triangles. So far as I know, he did not develop this version, nor add the continents, etc., unlike the triangulars. Here we see his excess nomenclature: three other titles (unused) besides Conformal.

Source: excerpt from a 36 x 57 cm (14 x 22") broadsheet. "The Butterfly Map"
at The Bancroft Library, University of California, Berkeley;
Xeroxed and spliced, then scanned and enlarged to 175% for a scale of 1/200 M by GK.
Cahill Conformal curvilinear graticule

As I've mentioned elsewhere, Cahill seemed to "steal" my ideas before I was born, including the "M" layout instead of his predominant Butterfly. Below is an unpublished Cahill draft, n.d., ca. 1927, of the Conformal variant in an "M" shape. I do not know the purpose of the internal scaffold lines, but the four diagonals across the triangles' altitudes once again show what I regard as his design mistake of dividing the meridian-perimeter geocells in half, e.g., at 22.5º W instead of 20º as I did (and Steve Waterman independently). Using 20º, the altitude of each octant would be a central meridian 45º from either edge, instead of a column of bisected geocells. But at least this draft has the far preferable 5º geocells.

Source: Xerox from Cahill collection at The Bancroft Library,
University of California, Berkeley;
scanned, spliced, and cleaned by GK.

Cahill Conformal variant, M-shape

The next item [1928] is similar to Cahill's double piece* in the April 1929 Monthly Weather Review, but not entirely.

* (as copied from my Cahill Resource):

1928-12-28 / 1929-04
Cahill, B.J.S., "Projections for World Maps"
—and text continued in separate pdf, plus illustrations:—
Cahill, B.J.S., "A New Map for Meteorologists: Equally Suitable for Small Areas, Continents, Hemispheres or the Entire World" – both from Monthly Weather Review, 57/4, 1929-04) p. 128-133; illus.
Has Cahill's only published [partial] world map with a one-degree graticule, except on land areas; as well, one of his only published five-degree world maps, regrettably discontinuous on two pages. See similar map in [1934] below, octants together, but in an awkward north-south spread, which I also show enlarged; and cf [1940] below, in Butterfly layout.

It is a broadsheet, 30 x 47 cm (12 x 18.5"), packed with tiny print because it was reduced from an original five times larger. [I have a complete scan of the small one, shown here, but only a Xerox-splice portion at the original size.]. So I re-enlarged the reduced typeface in separate frames below, to make its important content legible, especially the lower right section, "All Synoptic Charts from One Base Map", where Cahill again voices themes that I was discussing 45 years later, before I found he too had raised many of the same concerns.

The map itself is in still another layout, a North and South zigzag spread, which Cahill thought might be apropos for a consolidated world weather map.

Source: item #23 in The Bancroft Library Cahill Collection,
as scanned in several parts from original and spliced by GK in ClarisWorks Draw.

Note: whereas this map was split over two non-adjoining pages in the Monthly Weather Review, there was a single-page version in the 1934 "A World Map to End World Maps" article, but with different captions and 23 overlapping frames from various weather stations. Link is to image as scanned, rotated to horizontal, and enlarged by GK. The text from this broadsheet image will be enlarged in the subsequent figures.
Cahill, A New Map for Metorologists

Above is the map's broadsheet which I scanned and uploaded at its actual 1/5 size version.. In the next frame are detail-enlargements of its text portions. In the first subtitle cut below, we again see Cahill's confusing nomenclature: unillustrated mention of his otherwise unused terms "Authalic", "Orthometric". and "Graticular Scale". (I was also confused at first by the character "V", thinking it was a Roman numeral for "5" among Cahill's versions; but it is an abbreviation for Variant.) Note also the credit to Dr. Oscar S. Adams, the prominent mathematician whom Cahill hired to calculate all three Variants subsequent to the original.
Text detail, title

Below are more of the enlarged text details from the same broadsheet by Cahill; source as noted above. —GK
Text detail, pt.2
Text detail, pt. 3

Text detail, pt. 4
Text detail, pt. 5
Text detail, pt. 6
Text detail, pt. 7
Text, pt. 8
Below is the same text detail as the last piece above, but at its original size, as Xeroxed and spliced from the original at the Cahill Collection in the Bancroft Library, then scanned in two parts and spliced in ClarisWorks by GK.

Notice that the "E PLVRIBVS VNVM" sheets are more distinct in this original size image:
Cahill text, original size

Why am I focusing on the Conformal, when I like his original and the (inwardly-truncated) Equal-Area Variants better? For several reasons, shown separately below. (a) Apart from a big hand-colored wall-size draft of the Original, the Conformal was his most developed and even had a printed partial one-degree version of the North Atlantic (next image below); (b) In 1937, the International Meteorological Committee came within one vote of adopting the Cahill Conformal for world weather charting, on a single base map in place of five different ones being used; (c) In early 1975, when I was looking for a Cahill print to work on, (and thereby supplant my earlier use of Fuller's icosahedral), the 5º version in the Monthly Weather Review was the only practicable one then available to me; (d) But I didn't like the full-triangle version because one could not show complete polar areas, and I soon moved on to my own Cahill re-design, with a proportional geocell graticule.

(e) Then, in 2015, Jacob Rus began developing a Cahill-Keyes-style Conformal version within truncated triangles having 90º right-angle apexes. He has delved much more deeply into the math of the Conformal than I could or would. The results are promising so far, and shown at the end of this page. 

So here is the one-degree excerpt — but only an excerpt, because from what I saw at the B.J.S. Cahill collection at UC Berkeley, there was not an entire world map made at this resolution: only the better part of the northern and southern hemispheres on two sheets. A nice piece of work, and I wished the one-degree cells also covered land areas. Upon later examination, I saw that the northern latitude heights were larger than temperate ones: shades of Mercator, and there goes Greenland again: another reason why I strove to make my own Cahill, without great geocell disparities. Jacob Rus avoided the same pitfall.

Like Cahill's five-degree version in the broadsheet, this too was 1/20,000,000 in the original. (See next image after this.)
Source: Monthly Weather Review as indicated,
April 1929; jpeg screenshot of pdf, enlarged by GK
Cahill 1 degree, Monthly Weather Review

While at the Bancroft Library in 1983, I xeroxed a full-size 1/20,000 version, in many pieces, and later spliced them together as best I could. As mentioned, the map was not complete, but had significant portions of the northern and southern hemisphere on two sheets, and was dated Feb. 19, 1929. This is what a portion looks like.
Source: The Bancroft Library Cahill Collection. (Listing items 19 and 20.)
Cahill Conformal
              1 degree

Some Early Starts and Work in Progress based on the Conformal

In March 1975, eight years before I saw the one-degree Conformal, I began my effort to make a revised standard Cahill in the M-profile, by xeroxing the Monthly Weather Review illustrations, cutting their triangles out and re-configuring them. Then I enhanced the coastlines, added blue coloring, and spliced in an orthographic version of Antarctica, because the triangular Conformal could not piece together the frozen continent. (Note that I also drew in the missing triangle sides.)

This preliminary dummy, at 1/100 M, was a huge step forward for me, but already the shortcomings of the triangular Cahill Conformal were becoming apparent: the Antarctica problem, the half-geocell problem, and the uneven distribution of parallels, resulting in polar regions that were swollen as well as incomplete.

Whereupon, in November 1975 I drafted the first Cahill-Keyes with its proportional geocell graticule; coincidentally at the scale of 1/20 M used by Cahill to exhibit his Conformal. (My wall map dummy was at five degrees, to which I laboriously filled in one degree geocells by hand.)
Earliest C-K draft

I had thought the Conformal was unsuitable for a Cahill-Keyes master map, until 30 years later in 2015, when Jacob Rus sent me his aforementioned Conformal draft of Cahill-Keyes in truncated octants, enabling the reassembly of Antarctica, and an overall world map essentially similar to mine, but with better underlying math.

Below is is a screen grab, reduced to 10%, of a joint pdf of the Cahill-Keyes Beta 2 Megamap, and just below it, is a very early draft, 2015-04, of the Cahill-Keyes in a Conformal rendition by Jacob Rus, also at 1/1,000,000, set in the same grid as mine of 40,000 mm width. To see the full-size very sharp original pdf, download the combined pair here* and follow my instructions for easily viewing the humongous images, in the Foxit reader (now available for Windows, Mac, and Linux). *Note: you should "save link as", and then open it with the Foxit pdf reader. Suffice to say that my graticule is sui generis with proportional geocells, and that the Rus Conformal's math is over my head. But at minimum his produces better curves, and truly right-angle geocells, besides keeping my proportionality precept. (And Rus made the joint pdf, at a file size of only 18 MB, almost half of my single original.)
     [This one: the Gene Keyes Beta 2 version.]
Cahill-Keyes Megamap Beta 2, c/o Jacob Rus
     [Below: the Jacob Rus Conformal version]
Jacob Rus C-K Conformal 10%

Next is an early draft of a Jacob Rus Conformal a la Cahill-Keyes precepts, but without my grid overlay, and in a Butterfly profile. (And lacking an assembled Antarctica.) This too is a reduced version of a big image; the original jpeg is five times larger. Click on it once for full size, to reveal the one-degree graticule, hallmark of the Cahill-Keyes endeavor; reload to restore the small size.

Thus, another step in the slow saga of Cahill development. As the Danish poet Piet Hein put it, TTT: Things Take Time.
To view at full size, click image once. To restore small size, reload. Cahill-Keyes Conformal Butterfly by Jacob Rus

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