Let us now take the reverse problem, and, given the three cyclical projections, determine the shape. Let the _a c d_ projection be the Moenas of Urna, Ostrum, Bidens, Scena, Vestis. Let the _c d a_ be the Syces of Urna, Frenum, Plebs, Sypho, and the _d a c_ be the Alvus of Urna, Frenum, Uncus, Spicula. Now, from _a c d_ we have Urna, Frenum, Sector, Ostrum, Uncus, Ala, Bidens, Pallor, Cortis, Scena, Tergum, Aer, Vestis, Oliva, Tyro. From _c d a_ we have Urna, Ostrum, Comes, Frenum, Uncus, Spicula, Plebs, Pallor, Mora, Sypho, Tergum, Oliva. In order to see how these will modify each other, let us consider the _a c d_ solution as if it were a set of cubes in the _c d a_ arrangement. Here, those that go in the Arctos direction, go away from the plane of projection, and must be represented by the Syce of the cube in contact with the plane. Looking at the _a c d_ solution we write down (keeping those together which go away from the plane of projection): Urna and Ostrum, Frenum and Uncus, Sector and Ala, Bidens, Pallor, Cortis, Scena and Vestis, Tergum and Oliva, Aer and Tyro. Here we see that the whole _c d a_ face is filled up in the projection, as far as this solution is concerned. But in the _c d a_ solution we have only Syces of Urna, Frenum, Plebs, Sypho. These Syces only indicate the presence of a certain number of the cubes stated above as possible from the Moena projection, and those are Urna, Ostrum, Frenum, Uncus, Pallor, Tergum, Oliva. This is the result of a comparison of the Moena projection with the Syce projection. Now, writing these last named as they come in the _d a c_ projection, we have Urna, Ostrum, Frenum, Uncus and Pallor and Tergum, Oliva. And of these Ostrum Alvus is wanting in the _d a c_ projection as given above. Hence Ostrum will be wanting in the final shape, and we have as the final solution: Urna, Frenum, Uncus, Pallor, Tergum, Oliva. CHAPTER XI. A TESSARACTIC FIGURE AND ITS PROJECTIONS. We will now consider a fourth-dimensional shape composed of tessaracts, and the manner in which we can obtain a conception of it. The operation is precisely analogous to that described in chapter VI., by which a plane being could obtain a conception of solid shapes. It is only a little more difficult in that we have to deal with one dimension or direction more, and can only do so symbolically. We will assume the shape to consist of a certain number of the 81 tessaracts, whose names we have given on p. 168. Let it consist of the thirteen tessaracts: Urna, Moles, Plebs, Frenum, Pallor, Tessera, Cudo, Vitta, Cura, Penates, Polus, Orcus, Lacerta. Firstly, we will consider what appearances or projections these tessaracts will present to us according as the tessaractic set touches our space with its (_a_) Mala cubes, (_b_) Vesper cubes, (_c_) Pluvium cubes, or (_d_) Lar cubes. Secondly, we will treat the converse question, how the shape can be determined when the projections in each of those views are given.