[Written by Sir D'Arcy Wentworth Thompson. He probably wrote these instructions around 1897. The original copy of these instructions, correspondence about the machine, and the machine itself can be seen in the archives of the University of St. Andrews.]

In the propositions U, A, Y, I, we quantify respectively :- subject and predicate, subject alone, predicate alone, and lastly, neither. We may symbolize this fact by writing it :-

s | p |

U |

s | - |

A |

- | p |

Y |

- | - |

I |

+ | + |

U |

+ | - |

A |

- | + |

Y |

- | - |

I |

the corresponding negatives

s | p |

E |

s | - |

η |

- | p |

o |

- | - |

ω |

+ | + |

E |

+ | - |

η |

- | + |

o |

- | - |

ω |

In the these figures, subject and predicate are, in each premise, respectively :-

yx, zy, zx ; xy, zy, zx ; yx, yz, zx.

Now, by superimposing, one upon another, over these sets of symbols, – those, namely, for the propositions, for their unimplied qualifications, and for the figures, – we may ascertain the conclusion of any syllogism provided that we have in mind the following rules :-

- That two negative premises are inadmissible
- That if one premise be negative, the conclusion is negative.
- That the middle term must be quantified at least once.
- That nothing may be quantified in the conclusion that has not been quantified in the premises.
- That everything must be quantified in the conclusion that has been quantified in the premises :-

Except that, when the double-quantified affirmation proposition U occurs in conjunction with a less quantified proposition [viz. with A, Y, I, η, ο or ω] then the quantification of the former (other than its quantification of the middle term y) shall be neglected in the conclusion.

**Note.** This apparent exception simply depends upon the fact that in such cases the proposition U quantifies, or states, more than is required for the argument. __All y’s are all x’s__, when followed by the major proposition __all z’s are some y’s__, has stated more than is necessary, for the same conclusion, viz. that __all z’s are some x’s__, would follow equally from the major premise __all z’s are some x’s__. In other words, in the 1^{st} figure, UA leads only to the same result as AA, and in the 2^{nd} and 3^{rd} figures only to the same result as YA. In point of fact, the proposition U is only of real utility when used in conjunction with another doubly quantified proposition, viz. in UU, UE, perhaps even in the former case alone.

The method of superposition, indicated above, may be employed by itself, or used in the construction of a Reasoning Machine. The following examples illustrate the matter by itself.

- Given the combination AY
Then we have in the first figure, y x + - A

=z y - + Y

= I- - z x .. .. .. .. 2 ^{nd}..,x y + - A

=z y - + Y

= Y- + z x .. .. .. .. 3 ^{rd}..,y x + - A

=y z - + Y

= A+ - z x - Given the combination Aω
We have in the first figure, y x + - A

=z y - - ω

(neg.) = ω- - z x .. .. .. .. 2 ^{nd}..x y + - A

- Invalid,z y - - ω __y unquantified.__.. .. .. .. 3 ^{rd}..y x + - A

=y z - - ω

(neg.) = ω- - z x

Place the cards representing the propositions in their place and order on the base-board. The terms that are quantified will then appear through the holes.

- A red danger-signal in the first or lowest compartment shows the whole combination to be invalid (double negative).
- A red danger-signal in the compartment assigned to any figure shows that figure to be invalid (unquantified middle).
- Quantify in the conclusion anything that is shown to be quantified in the premises [In other words, transfer to the zx of the conclusion the symbols (+ or -) that the z and x bear in the premises.], unless, a green signal of caution directs [according to the preceding tule V, EXC.] that the quantification of U is to be disregarded in the conclusion.