76 
GENETICS: W. E. CASTLE 
Proc. N. a. S. 
of reduction to a curved line lying in one plane." Obviously the arrange- 
ment approaches the linear, but it is curvilinear rather than rectilinear 
as shown in the chromosome maps. But if we grant that the arrangement 
is in any sense linear, then it must be granted also that double and triple 
crossing-over are likely to occur, indeed we cannot in that case reasonably 
explain the observed regrouping of characters on any other assumptions. 
But if double crossing-over occurs of tenest when long distances are involved, 
it follows that the long distances will have been apparently shortened. 
If now in the model based on Gowen's observations, we lengthen the long 
distances by counting twice each assumed double cross-over and thrice 
each assumed triple cross-over, we do in fact obtain a distribution of the 
eight genes substantially in a rectilinear hand. This strongly supports the 
linear hypothesis. Yet there remains the serious logical difficulty that 
the long distances were already long enough before we increased them 
by allowing for double crossing-over. When so increased they will of 
necessity be too long, since they will now exceed 50, the limiting value for 
cross-over percentages involving linkage. It must then be frankly rec- 
ognized that map distances do not correspond with cross-over percentages 
(although they are based on them) except when the cross-over values are 
very low. This has been made clear by Haldane^ who designates the map 
units "morgans" to distinguish them from the directly observed cross- 
over percentages. He gives a useful table for converting cross-over per- 
centages into map-distances and vice versa. How widely the two may di- 
verge is illustrated in the following table in which are shown the linkage 
relations of the gene, star, with other second chromosome genes. 
OBSERVED CROSS-OVER 
PERCENTAGE (TABLE 
140, bridges and 
morgan) 
LINEAR MAP distance 
(fig. 72, BRIDGES AND 
morgan) 
Star — streak 
15.9 
15.4 
Star — dachs 
Not given 
29.0 
Star— black 
37.9 
46 . 5 
Star — purple 
43.7 
52.7 
Star — vestigial 
43.3 
65.0 
Star — curved 
45.9 
73.5 
Star — plexus 
46.7 
96.2 
Star — arc 
Not given 
98.4 
Star — speck 
Not given 
105.1 
Star — morula 
Not given 
106.3 
It appears from the table that cross-over percentages, compared with 
map distances, increase very slowly after they have once become as high 
as 40. According to Haldane's table, the cross-over percentage would 
not attain 50 until the map-distance reaches infinity. 
