“First, let us suppose that the above [Biliteral] Diagram is an enclosure assigned to a certain Class of Things, which we have selected as our ‘Universe of Discourse,’ or, more briefly, as our ‘Univ’.

“Secondly, let us suppose that we have selected a certain Adjunct, which we may call x, and have divided the large Class, to which we have assigned the whole Diagram, into the two smaller Classes whose Differentiæ are x and not-x (which we may call x’), and that we have assigned the North Half of the Diagram to the one (which we may call… the x-Class), and the South Half to the other (which we may call… the x’-Class).

“Thirdly, let us suppose that we have selected another Adjunct, which we may call y, and have subdivided the x-Class into the two Classes whose Differentiæ are y and not-y, and that we have assigned the North-West Cell to the one (which we may call the xy-Class) and the North-East Cell to the other (which we may call the xy’-Class).

“Fourthly, let us suppose that we have subdivided the x’-Class in the same manner, and have assigned the South-West Cell to the x’y-Class, and the South-East Cell to the x’y’-Class” (p22-23).

For example, if x means ‘old’, so that x’ means ‘new’, and if y means ‘English’, so that y’ means ‘foreign’, we have the following diagram:

“Let us agree that a Red Counter [represented below as a dotted circle or as an 'I'], placed within a Cell, shall mean ‘This Cell is occupied‘ (i.e. ‘There is at least one Thing in it’).

“Let us also agree that a Red Counter, placed on the partition between the two Cells, shall mean ‘The Compartment, made up of these two Cells, is occupied; but it is not known whereabouts, in it, its occupants are.’ Hence it may be understood to mean ‘At least one of these two Cells is occupied: possibly both are.

“Let us also agree that a Grey Counter [represented below as a hollow circle or as an 'O'], placed within a Cell, shall mean ‘This Cell is empty‘ (i.e. ‘There is nothing in it’)” (p26).

By this formula we can visually represent the following propositions (p34, 35):

“The Reader should now get his genial friend to question him, severely, on [the above] two Tables” (p34).

Let us now support, firstly, “that we change [the Biliteral Diagram] into a Triliteral Diagram by drawing an Inner Square, so as to divide each of its 4 Cells into 2 portions, thus making 8 Cells altogether” (p39).

“Secondly, let us suppose that we have selected a certain Adjunct, which we may call m, and have subdivided the xy-Class into the two classes whose Differentiæ are m and m’, and that we have assigned the N.W. Inner Cell to the one (which we may call… the xym-Class)” (p40) and that we have subdivided the remaining classes in the same manner. “It is evident that we have now assigned the Inner Square to the m-Class, and the Outer Border to the m’-Class” (p40).

Thus we can visually represent propositions of the form (p49):

It is now possible to draw two propositions on the same diagram, one in terms of x and m, the other in terms of m and y, and to visually derive a third proposition in terms of x and y. Consider, for example, the following pair of propositions:

“Some, who deserve the fair, get their deserts;
None but the brave deserve the fair” (p101).

If the “Univ is ‘persons’; m = persons who deserve the fair; x = persons who get their deserts; y = brave” (p140), then the Triliteral Diagram represents that “some m are x [and] no y’ are m”:

The Biliteral Diagram to the right, relating x and y, can be derived from the Triliteral Diagram according to the following procedure: for each quarter of the Triliteral Diagram, “if it contains a ‘I’ in either Cell, it is certainly occupied, and you may mark the… [corresponding] quarter of the Biliteral Diagram with a ‘I’. If it contains two ‘O’s, one in each Cell, it is certainly empty, and you may mark the… Biliteral Diagram with a ‘O’” (p53).

Therefore the derived Biliteral Diagram above concludes that “some brave persons get their deserts” (p140).

This method may also be used to validate the correctness, or expose the fallaciousness, of a proposed syllogism. For example:

“Some epicures are ungenerous;
All my uncles are generous
[Therefore] My uncles are not epicures” (107).

If “Univ. ‘persons’; m = generous ; x = epicures ; y = my uncles” (p145):

“Hence [the] proposed Conclusion is wrong, the right one being ‘Some epicures are not uncles of mine’” (p145).

Writes Carroll, “Mental recreation is a thing that we all of us need for our mental health; and you may get much healthy enjoyment, no doubt, from Games, such as Back-gammon, Chess, and new Game ‘Halma’. But, after all, when you have made yourself a first-rate player at any one of these Games, you have nothing real to show for it, as a result! You enjoyed the Game, and the victory, no doubt, at the time: but you have no result that you can treasure up and get real good out of. And, all the while, you have been leaving unexplored a perfect mine of wealth. Once master the machinery of Symbolic Logic, and you have a mental occupation always at hand, of absorbing interest, and one that will be of real use to you in any subject you may take up. It will give you clearness of thought — the ability to see your way through a puzzle — the habit of arranging your ideas in an orderly and get-at-able form — and, more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will so continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art. Try it. That is all I ask of you!” (pXVII).

The reader may pursue the outcome of the following two propositions on his or her own time:

“Nothing intelligible ever puzzles me;
Logic puzzles me” (p102).

Philosophical Geometry contrasts with axiomatic geometry: the latter “discipline is founded on a group of propositions considered self-evident or necessary, from which a chain of further propositions can be deduced” (p3), whereas Philosophical Geometry is a “property of mind in general and not a specialty of the analytic mind… For example, in ‘Meno’, Plato shows geometry as a birthright of mind in general: Meno’s slave, unhampered by his lack of background, comprehends a geometric necessity with which Socrates confronts him” (p4).

Philosophical Geometry consists of theoria and practica. “Theoria is the adequate expression of geometric experience. As such, it is dependent upon the perception of the geometer; it is a subject… Practica is a necessary structure of two-dimensional events and is independent of the geometer’s perception. It is the object of the discipline… The aim of Philosophical Geometry is the individual elaboration of theoria… Theoria depends upon practica through the perception of the geometer, and it is due to the variance in perception that differences in practica occur” (p12).

Therefore, geometry is the product of geometric gesture.

“On a homogeneous two-dimensional field best termed a plane, the stylus is posited in a gesture of inscription. The contact of stylus with plane breaks the homogeneity of the undifferentiated surface into a heterogeneity of the point of contact and the remainder of the plane… Whatever the final complexity of inscription, the [practica] must pass through this initial stage: the contact of stylus with plane” (p17).

“Practica can progress beyond the point only by motion of the stylus. Motion of the stylus produces an inscription best termed a line” (p17-8).

“To qualify the line as a two-dimensional element is a contradiction in terms. A line cannot be a surface. And yet, contrary to the point, it cannot be considered an ideal indivisible marking. Although, as line, it cannot be a surface, it clings to the plane by a dimension of length and thus forces upon contemplation a second dimension of width which is the thickness of the inscripting stylus” (p18).

“To examine the line, motion of stylus has to be arrested… If the motion of stylus is not arrested, a continuous line results. This uniform, indefinitely prolonged motion of stylus finds its perfect representation only in the circle” (p18-9).

“The maximum number of pieces into which a pancake [or pie] can be cut with 6 slices” is 22 (p78):

22 is also the 4th pentagonal number — whereas “the 4th centered hexagonal number, obtained by arranging hexagonal layers around a central point,” is 37 (p99):

“By a different division [of the above] the nth centered hexagonal number is equal to 6T_n-1 + 1, where T_n is the nth triangular number” (p100). Here we see a composition using the 3rd triangular number:

And the 4th triangular number? — 10, the tetractys, a figuration “so holy [to the Pythagoreans] that they even swore oaths by it” (p61, c.f. the obverse seal):