Because it's almost always impossible to use a circular Venn diagram to show correct - proportional - overlaps between three or four sets (and more), I'll suggest something a little different.

I came up with something I call an "Eulergrid" which shows a bar graph, where each bar is an element in the power set of intersected sets, and a grid of overlap cases underneath (e.g., for three sets: A, B, C, A ∩ B, B ∩ C, A ∩ C, A ∩ B ∩ C).

The bar graph shows the overlap cardinalities between set intersections contained in the power set. The grid shows the intersection between one and more sets, and is aligned to the value shown in the bar graph column. The bar graph is sorted by overlap cardinality, presented from left to right, from least to greatest cardinality. (I leave out visualizing the empty set, although strictly speaking this is also a valid subset.)

While an Eulergrid is admittedly less intuitive to read, at first, than a circular Venn diagram, it can always show all true, proportional overlaps between all the sets, and without adding distortion or visual errors from "impossible" Venn overlaps.

The R script used to make Eulergrids will scale up to however many sets you need to show intersections for, but it will create an exponentially-wider figure as the total number of permutations of intersections increase as a power of 2 (three sets have eight power set subsets, intersections of four sets have sixteen subsets; five sets have thirty-two subsets, etc.).

To demonstrate, here's an example of what an Eulergrid figure looks like:

The green denotes the count for that subset. Yellow coloring, in the context of this figure, represents cell-specific cardinality, i.e. the counts that are unique to a single cell type or dataset.

As a way to read this, for example, 42% of the total element overlaps over these five cells types involve SKNSH in some way. Of all those overlaps, roughly half can be assigned to SKNSH alone.

Here's the R code for plotEulergrid.R:

Here's a Perl-based wrapper to this R script, called eulergrid.pl:

Here's an example of calling the Perl wrapper on the command line, which was used to make the figure shown above:

$ ./eulergrid.pl \
    --setNames=GM06990,HepG2,K562,SKNSH,TH1 \
    --plotTitle="Footprint__overlaps__for__multiple__cell__lines\n(FDR__0.001)" \
    --setCardinalities=212350,233552,270586,287731,240701,93351,64049,89860,110579,62852,96806,89476,62075,64644,90129,30893,51178,53416,29083,32041,51033,28922,28279,48629,27407,22805,23548,39400,22418,21029,17172 \
    --setTotal=689952 \
    --outputFilename=results/footprintOverlaps/overlaps.fdr0p001.112409.png \
    --offCellColor="gray80" \
    --onCellColor="springgreen4" \
    --ctsCounts=65897,97624,173336,150753,91965

The option --ctsCounts refers to the yellow coloring I describe up above, representing "cell-type-specific" counts.

The option --setCardinalities shows the counts of sets and intersections of sets: A, B, C, D, A ∩ B, A ∩ C, A ∩ D, B ∩ C etc.

Hopefully, this gives you some ideas or at least an understanding that Venn diagrams cannot always represent intersections between more than three sets (and usually not even between three sets).

EDIT: My Eulergrid idea seems to have been turned into UpsetR, without attribution. Ah well. In any case, I have reloaded the demo image to demonstrate the initial visual premise.

You say proportional Euler diagram with four sets, but that's an impossibility in the general case (try sketching it proportionally). You can make a simple 4-way Venn pretty easily with a few different packages, here's an example using venn from the gplots package:

library(gplots)
test1 <- c("dog", "cat", "monkey", "fish", "cow", "frog")
test2 <- c("cat", "frog", "aardvark", "monkey", "cow", "lizard", "bison", "goat")
test3 <- c("whale", "cat", "cow", "dog", "worm") 
test4 <- c("dog", "bird", "plant", "fly", "cow", "horse", "goat")

venn(list(A=test1,B=test2,C=test3,D=test4))

I know this is an old post, but for posterity: UpSetR is an R implementation of "UpSet: Visualization of Intersecting Sets".

It generates plots like this:

Below is an image created using the EulerView plugin from Tulip. There are no splits of the data. I think its quite quick to see which items are unique to one set and which items are shared between many/all.

Ok. For-dummy 2-steps tutorial is here.

Step 1: Upload a data table like this

Step 2: Drag everything to Set(s)

It is an area-proportional Euler diagram. You can hover your mouse over to get info of the intersections. I made it with this tool. Nice graphics and customization, too. Too bad I couldn't turn off the area-proportional thing but it's worth a try.

If you want a little bit less easy, try the function vennCounts and vennDiagram from limma in R. I found a really good example here. You can start with a data.frame similar to the one I made in the first picture.

There is a really useful handy tool available called Venny. Below I have created the fourway venn diagram with your data.

http://bioinfo.genotoul.fr/jvenn/

jvenn displays up to six sets using classical and Edwards-Venn layouts. It works via a web browser and can output as PNG or CSV text for interrogating the overlaps, etc.

One new solution is EulerForce (Force-directed layout for Euler diagrams): http://kar.kent.ac.uk/41437/1/2014_JVLC_eulerForce.pdf

http://www.eulerdiagrams.org/eulerForce/

This is how the text file you edit looks like:

DIAGRAM

ABSTRACTDESCRIPTION
0 b c d ac bc cd abc

CONTOURS
a|562|343|530|341|470|335|482|275|498|255|566|335|
b|610|355|602|355|561|350|579|353|482|359|555|409|616|409|677|409|677|366|677|323|644|319|644|291|645|261|542|275|514|295|506|335|546|339|590|341|
c|452|335|476|338|524|344|578|351|594|351|615|347|685|303|685|247|686|188|604|165|543|165|482|165|420|165|409|287|450|255|410|335|466|339|438|335|464|275|
d|418|295|361|347|385|341|458|341|498|343|442|335|458|275|

And you can see in either link how the resulting figure looks like