ANGIOSPERMS. 419 



found was the case in Dictammts Fraxinella (Fig. 305) ; it is shown in the diagram in 



Fig. 349, where the later formed stamens are not black like the first ones but only 



shaded. He considers himself entitled to conclude from Payer's figures and descriptions 



that the same proceeding takes place in the nearly allied Ruta and in the families of the 



Oxalideae, Zygophylleae, and Geraniaceae which belong to the same cycle of affinity, 



and that in them also five stamens are subsequently interposed between the five calycine 



stamens which were first formed. If we suppose the five interposed stamens removed, 



there remains a regular pentamerous flower with four whorls of five alternating members, 



such as is seen in the nearly related Lineae and Balsamineae. Whether the new stamens 



arise at the same height as those first formed (diplostemony) or lower than they, evidently 



depends on where more space is left free by the changes in the form of the growing 



torus. We have just seen instances of such interposition according 



to the conditions of space in the torus in the Rosaceae. But other o 



facts of development are opposed to this explanation in the case 



of obdiplostemonous flowers x . According to Frank there can be 



no interposition here, but the corolline stamens are the older, and 



the calycine are formed afterwards in alternation with them. 



However this may be, it must be borne in mind that the normal 



alternation of the parts of the flower is only a fact of experience, 



which loses its validity as a general rule as soon as a number of 



opposing facts are known. FIG. 349- Fiorai dia- 



Floral formulae. The diagram may if necessary be partially ^^^ m ^ Ffmxf ' 

 replaced by a formula composed of letters and figures ; in such a 

 floral formula the relative positions cannot always be exactly given, but it has the 

 advantage of being able to be expressed in ordinary type, and, which is perhaps 

 of greater importance, it is capable of a wider generalisation since the figures may 

 be replaced by letters as numerical coefficients. 



The construction and application of such formulae will be easily made intelligible by 

 a few examples 2 . 



The formula 6*3 P^ St 3 4- 3 C^ corresponds to the diagram of the Liliaceae (Fig. 340), 

 and means therefore that each of the two perianth-whorls, the outer of sepals S and the 

 inner of petals P, consists of three members ; that the androecium St consists of two 

 whorls of three members (3 + 3) and the gynaeceum C of one of the same kind ; the 

 diagram shows that these whorls of three members alternate without interruption, but 

 as this is the normal case in flowers, it is not specially indicated. The formula 

 63 /*3 5/3 2 + 3 C3 + 3 gives the numerical relations of the flower of Butomus umbellatus 

 (Fig. 299) ; it differs from the previous one in the circumstance that the gynaeceum C 

 consists of two whorls of three carpels each (3 + 3), and that in the androecium St the 

 typical three stamens of the outer circle are replaced by two stamens, as is expressed by 

 the sign 3 2 . The formula So P$ ^3 + 3 3 answers to the diagram of the flower of 

 Bambusa and differs from the first only in having So instead of 6*3, which means that 

 the outer perianth whorl is absent. The numerical relations of the flower of the Orchi- 

 deae (Fig. 344 A) would be expressed by the formula 6*3 P$ St'i + o 3, in which the symbol 

 St'i + o signifies that all the members of the inner whorl of the androecium are sup- 

 pressed, but that only the two obliquely posterior stamens are wanting in the outer whorl, 

 the anterior one being fully developed ; the position of the two dots above the number i 

 means that the abortive members are the posterior ones ; if they were anterior the dots 

 would be placed beneath the number as in the formula SoPoSf^ + o 2, which 



1 Frank, Ueber d. Entw. einiger Bliithen mit besoriderer Beriicksichtigung d. Theorie d. Inter- 

 ponirung, in Pringsheim's Jahrb. f. wiss. Bot. Bd. X. p. 20. 



" Grisebach (Grundriss d. syst. Bot. Gottingen, 1854) depicted the numerical relations of the parts 

 of the flower in this way, by simply writing the numbers of the alternating members one after 

 another, and indicating cohesions by strokes. 



E e 2 



