Therefore = Es ist daher The sentence only becomes complete by reading the = sign in the formula as gleich (“equals”). The equation presented here seems to be faulty, though. There are basically three possible interpretations. First, it may be that Mendel is asking his readers to treat the equation literally as an algebraic expression (some of his language in the preceding seems to imply as much; see p. 30, s. 2). But all one can get from correct transformations of the expression on the left hand side of the equation is the formula AA + 2Aa + aa (as later Mendelians like de Vries and Correns also had it), not A + 2Aa + a. There only seems to be one solution for the equation then, namely if one assumes that A = a = 1 (only in this case will it hold that AA = A and aa = a). This, however, does not make any sense, since A and a are supposed to be different. The second possible interpretation assumes that Mendel used fraction notation in order to derive the combinatorial series (see p. 21, s. 2). In this case, A and a are labels or indices for different factors only, not variables, and the signs for arithmetic operations like addition or division are not to be read arithmetically; Mendel might as well have written something like {(A,A), (A,a), (a,A), (a,a)} = {(A), 2(A,a), (a)} (we thank Michel Durinx for important clarifications on this point). Joseph Salomon, author of a popular textbook and a teacher of Mendel, warned his readers explicitly in his presentation of combinatorial analysis not to mistake his notation for an arithmetic one (Joseph Salomon, Lehrbuch der Elementar-Mathematik für Ober-Realschulen, Vienna: Carl Gerold und Sohn, 1853, p. 152). Since Mendel, unlike modern geneticists, believed anyway that factors of the same kind form an intimate union, whereas hybrid combinations remain in tension (see p. 41, s. 6), he simply did not bother to write out combinations composed of the same factors, denoting them by one letter only (see Robert C. Olby, “Mendel No Mendelian?” History of Science 17 (1979), pp. 53–72). Finally, the third option is that the equation may not have been meant to be an equation at all in any formal sense, as indicated by the fact that Mendel uses “or” instead of the equation sign later on in similar circumstances (see p. 31, s. 7). According to this third interpretation, the equation simply correlates possible combinations of gametes on the right hand side with their phenotypic outcomes on the left hand side (Vítezslav Orel and Daniel L. Hartl, “Controversies in the Interpretation of Mendels Discovery”, History and Philosophy of Life Sciences 16 (1994), pp. 436–455). Indeed, the preceding paragraph does use A, a, and Aa to designate the “products” of combinations of different kinds of pollen and germ cells, the latter being presented in “the form of a fraction”. A and a would in this case designate quite different things on the two sides of the equations: gametes of different kind on the left hand side, and plants of different kind on the right hand side. Whichever of the three interpretations one adopts, it seems quite clear that Mendel is closely entangling natural language, mathematical operations, and a particular understanding of underlying biological processes in this crucial passage of his 1866 paper.
Therefore = Es ist daher The sentence only becomes complete by reading the = sign in the formula as gleich (“equals”). The equation presented here seems to be faulty, though. There are basically three possible interpretations. First, it may be that Mendel is asking his readers to treat the equation literally as an algebraic expression (some of his language in the preceding seems to imply as much; see p. 30, s. 2). But all one can get from correct transformations of the expression on the left hand side of the equation is the formula AA + 2Aa + aa (as later Mendelians like de Vries and Correns also had it), not A + 2Aa + a. There only seems to be one solution for the equation then, namely if one assumes that A = a = 1 (only in this case will it hold that AA = A and aa = a). This, however, does not make any sense, since A and a are supposed to be different. The second possible interpretation assumes that Mendel used fraction notation in order to derive the combinatorial series (see p. 21, s. 2). In this case, A and a are labels or indices for different factors only, not variables, and the signs for arithmetic operations like addition or division are not to be read arithmetically; Mendel might as well have written something like {(A,A), (A,a), (a,A), (a,a)} = {(A), 2(A,a), (a)} (we thank Michel Durinx for important clarifications on this point). Joseph Salomon, author of a popular textbook and a teacher of Mendel, warned his readers explicitly in his presentation of combinatorial analysis not to mistake his notation for an arithmetic one (Joseph Salomon, Lehrbuch der Elementar-Mathematik für Ober-Realschulen, Vienna: Carl Gerold und Sohn, 1853, p. 152). Since Mendel, unlike modern geneticists, believed anyway that factors of the same kind form an intimate union, whereas hybrid combinations remain in tension (see p. 41, s. 6), he simply did not bother to write out combinations composed of the same factors, denoting them by one letter only (see Robert C. Olby, “Mendel No Mendelian?” History of Science 17 (1979), pp. 53–72). Finally, the third option is that the equation may not have been meant to be an equation at all in any formal sense, as indicated by the fact that Mendel uses “or” instead of the equation sign later on in similar circumstances (see p. 31, s. 7). According to this third interpretation, the equation simply correlates possible combinations of gametes on the right hand side with their phenotypic outcomes on the left hand side (Vítezslav Orel and Daniel L. Hartl, “Controversies in the Interpretation of Mendels Discovery”, History and Philosophy of Life Sciences 16 (1994), pp. 436–455). Indeed, the preceding paragraph does use A, a, and Aa to designate the “products” of combinations of different kinds of pollen and germ cells, the latter being presented in “the form of a fraction”. A and a would in this case designate quite different things on the two sides of the equations: gametes of different kind on the left hand side, and plants of different kind on the right hand side. Whichever of the three interpretations one adopts, it seems quite clear that Mendel is closely entangling natural language, mathematical operations, and a particular understanding of underlying biological processes in this crucial passage of his 1866 paper.