CHAPTER X.

CYCLICAL PROJECTIONS.

Let us denote the original position of the cube, that wherein Arctos goes Z, Cuspis X, and Dos Y, by the expression,

Z X Y (1) _a_ _c_ _d_

If the cube be turned round Cuspis, Dos goes [=Z], Cuspis remains unchanged, and Arctos goes Y, and we have the position,

Z X Y _[=d]_ _c_ _a_

where

Z _[=d]_

means that Dos runs in the negative direction of the Z axis from the point where the axes intersect. We might write

[=Z] _d_

but it is preferable to write

Z _[=d]_.

If we next turn the cube round the line, which runs Y, that is, round Arctos, we obtain the position,

Z X Y (2) _c_ _d_ _a_

and by means of this double turn we have put _c_ and _d_ in the places of _a_ and _c_. Moreover, we have no negative directions. This position we call simply _c d a_. If from it we turn the cube round _a_, which runs Y, we get

Z X Y _d_ _[=c]_ _a_,

and if, then, we turn it round Dos we get

Z X Y _d_ _a_ _c_

or simply _d a c_. This last is another position in which all the lines are positive, and the projections, instead of lying in different quadrants, will be contained in one.

The arrangement of cubes in _a c d_ we know. That in _c d a_ is:

{ Vestis Oliva Tyro Third { Scena Tergum Aer Floor. { Saltus Sypho Remus

{ Tibicen Mora Merces Second { Bidens Pallor Cortis Floor. { Moles Plebs Hama

{ Comes Spicula Mars First { Ostrum Uncus Ala Floor. { Urna Frenum Sector

It will be found that learning the cubes in this position gives a great advantage, for thereby the axes of the cube become dissociated with particular directions in space.

The _d a c_ position gives the following arrangement:

Remus Aer Tyro Hama Cortis Merces Sector Ala Mars

Sypho Tergum Oliva Plebs Pallor Mora Frenum Uncus Spicula

Saltus Scena Vestis Moles Bidens Tibicen Urna Ostrum Comes

The sides, which touch the vertical plane in the first position, are respectively, in _a c d_ Moena, in _c d a_ Syce, in _d a c_ Alvus.

Take the shape Urna, Ostrum, Moles, Saltus, Scena, Sypho, Remus, Aer, Tyro. This gives in _a c d_ the projection: Urna Moena, Ostrum Moena, Moles Moena, Saltus Moena, Scena Moena, Vestis Moena. (If the different positions of the cube are not well known, it is best to have a list of the names before one, but in every case the block should also be built, as well as the names used.) The same shape in the position _c d a_ is, of course, expressed in the same words, but it has a different appearance. The front face consists of the Syces of

Saltus Sypho Remus Moles Plebs Hama Urna Frenum Sector

And taking the shape we find we have Urna, and we know that Ostrum lies behind Urna, and does not come in; next we have Moles, Saltus, and we know that Scena lies behind Saltus and does not come in; lastly, we have Sypho and Remus, and we know that Aer and Tyro are in the Y direction from Remus, and so do not come in. Hence, altogether the projection will consist only of the Syces of Urna, Moles, Saltus, Sypho, and Remus.

Next, taking the position _d a c_, the cubes in the front face have their Alvus sides against the plane, and are:

Sector Ala Mars Frenum Uncus Spicula Urna Ostrum Comes

And, taking the shape, we find Urna, Ostrum; Moles and Saltus are hidden by Urna, Scena is behind Ostrum, Sypho gives Frenum, Remus gives Sector, Aer gives Ala, and Tyro gives Mars. All these are Alvus sides.

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.