According to the Brefield system, as summarized by his colleague Dr. Von Tavel, Fungi are divided into two primary classes: (1) the _Phycomycetes,_ or lower fungi nearest like the algae, _consisting of a one-celled thallus with sexual as well as non-sexual modes of reproduction_, and (2) the Mesomycetes and the Mycomycetes, _having a divided or many celled thallus, propagated by non-sexually formed spores_. The Phycomycetes are further divided into two large sections, based on their methods of reproduction, termed, respectively, Zygomycetes and Oomycetes. These include the old typical Mucors, the Peronosporeae or "rotting moulds," once classed with the Hyphomycetes, the Saprolegniaceae, "Fish Moulds," of aquatic habit, the Entomophthoraceae, "Insect Moulds," together with some minor groups. The Mesomycetes connect the Phycomycetes with the Mycomycetes. The class Mycomycetes is primarily divided into two sections, viz., Ascomycetes and Basidiomycetes, with the Ustilagineae, "Smut Fungi," in Mesomycetes, forming a transitional group between Phycomycetes and the Basidiomycetal group of the higher fungi. The Ascomycetes are primarily subdivided into _Exoasci_ and _Carpoasci,_ groups based on the character of the asci. In the first, _Exoasci,_ the asci are naked and borne directly on the mycelium; in the second, _Carpoasci,_ they are enclosed in a wrapper composed of fertile hyphae and sterile threads, having also accessory fruit forms. The first includes Endomycetes and Taphrineae. In the second are included the groups Gymnoasci, Perisporaceae, Pyrenomycetes, Hysteriaceae, Discomycetes, and Helvellaceae. The Basidiomycetes characterized by the possession of basidia are arranged in two groups, based on the character of the basidia: (1) the Protobasidiomycetes, in which the basidia are septate, divided, and (2) the Autobasidiomycetes, in which the basidia are not divided, and bear a definite number of spores. The first of these (Protobasidiomycetes) includes the following distinct groups: (1) the Uredineae, "Rust Fungi," which have horizontally divided basidia, always free, never enclosed; (2) the Auricularieae, having basidia somewhat resembling those of the Uredineae, but which are borne in fruit bodies with open hymenia; (3) Pileacreae, having horizontal septate basidia in closed receptacles; and (4) Tremellineae, having vertically divided basidia borne in gymnocarpous receptacles--that is, those in which the hymenium is exposed while the spores are growing. The Autobasidiomycetes are characterized by undivided basidia, bearing spores only at the apex. This group is subdivided into three sections: (1) Dacryomycetes, which includes the lowest of the Tremelloid forms, with club-shaped basidia, nearly approaching the true Hymenomycetal type, together with several groups of minor import; (2) Gasteromycetes; and (3) Hymenomycetes, with Phalloideae placed in the group as a subsection of Gasteromycetes.