LINKAGE AND CROSSING-OVER 443 
greater than the cross-over percentage between A and C, and the dis- 
crepancy increases with the magnitude of the values involved. This 
fact has been accounted for in two different ways. First, it may be 
supposed that the arrangement of the genes is really not linear, that 
B lies out of line with A and C, so that AC will be less than the sum of 
AB and BC, and that the more distant genes are no farther apart than 
indicated by the cross-over percentages between them. This expla- 
nation has met with more difficulties than it has cleared away. The 
second explanation is that the map-distances indicate proportionate 
numbers of breaks in the linkage chain between points, not propor- 
tionate numbers of changes of relation between genes at particular 
points. Thus, suppose genes ABCDE of a linkage system meet their 
allelomorphs, abcde, in a cross and gametes are later formed by the 
cross-bred as follows, (1) ABcde, (2) ABcdE, and (3) AbcDe. Assum- 
ing that the arrangement is linear, we must suppose that one break 
in the linkage chain has occurred in (1), two breaks in (2), and three 
breaks in (3). But if we did not have genes BCD under observation, 
and merely noted the relation of A to E, we should infer that in case 
(x) and in case (3) a single crossover had occurred. We should on that 
basis underestimate the amount of breaking in the linkage chain. 
Accordingly the construction of maps on the basis of short distances 
summated is justifiable, provided the arrangement is linear, as it seems 
to be. But it must be borne in mind that the map distances do not 
correspond with cross-over percentages (although they are based on 
them) except in the case of very short distances. Map distances often 
exceed 50, but cross-over percentages cannot do so, as already pointed 
out. To get a distinctive name for the map units, Haldane has called 
them units of Morgan or simply “morgans.”’ Haldane has computed 
a formula for converting cross-over percentages into “‘morgans”’ and 
vice versa. He finds that the two correspond only for very low values 
(5 or less) and diverge more and more as the observed cross-over per- 
centages approach 50. Haldane’s formula may be stated thus. If 
three genes, A, B, and C, occur in a common linkage group, and the 
cross-over percentages are known between A and B and between B and 
C, we may predict with a probable error of not over 2 per cent, what 
cross-over percentage will be found to occur between Aand C. Call- 
ing the cross-over percentage between A and B, m, and that between 
B and C, n, the cross-over percentage between A and C will lie between 
(m-+n) and (m+n—2mn). It will approach the former for amounts 
of 5 or less, and the latter for amounts of 45 or over. Ina useful table 
