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The special rules in the first figure are two; namely—

1. The minor premiss must be affirmative.

2. The major premiss must be universal.

In the second figure the following rules apply:

1. One of the premises must be negative.

2. The major premiss must be universal.

In the third figure the special rules are1. The minor premiss must be affirmative. 2. The conclusion must be particular.

In the fourth figure there are three special rules; namely— 1. If the major premiss be affirmative, the minor must be universal. 2. If the minor premiss be affirmative, the conclusion must be particular.

3. If either of the premises be negative, the major must be universal. By the application of these rules, the legitimate moods are · reduced to six in each figure, or twenty-four in all; and of these five are rejected as useless, because they give only a particular conclusion when a universal one might have been drawn from the premises.

In order to impress upon the memory the nineteen allowable moods of the syllogism, logicians have contrived the following mnemonic lines, in which the three vowels denote the quantity and quality of the major premiss, the minor premiss, and the conclusion respectively. The consonants have also a meaning, which will be presently shown.

Fig. 1. Barbara, celarent, darii, ferioque prioris.

Fig. 2. Cesare, camestres, festino, baroko, secundæ.

Fig. 3. Tertia, darapti, disamis, datisi, felapton, bokardo, feriso, { habet; quarta insuper addit.

Fig. 4. `Bramantíp, camenes, dimaris, fesapo, fresison.

By way of illustration we shall give an example in each figure. Bar Every (plane figure) is bounded by lines;

ba Every triangle Is (a plane figure);

ra therefore; Every triangle is bounded by lines.

Ces No circle is (bounded by straight lines);

a Every square is (bounded by straight lines);
re therefore; No square is a circle.

Da Every (square) is a parallelogram; ·

rap Every (square) is an equilateral figure;

ti therefore; Some equilateral figures ARE parallelograms.

Bram Every triangle is a (plane figure);

an Every (plane figure) is bounded by lines;

tip therefore; Some figures bounded by lines ARE triangles.

The doctrine of the syllogism may also be derived from the maxim of Aristotle, termed "dictum de omni et nullo," and which may be expressed as follows; namely, whatever may be universally affirmed or denied of any universal term, may be affirmed or denied of everything contained under it. Now it is only to moods in the first figure that this principle can be directly and at once applied, and therefore this figure has been termed perfect, and the other figures which require to be reduced to the first before they can be compared with the dictum are termed imperfect.

The reduction of a syllogism is either ostensive or ad impossibile. Ostensive reduction consists in obtaining in the first figure either the same conclusion, or one illatively convertible into the same conclusion, by either transposing the premises of the original proposition so that the major may become the minor, and vice versâ, or illatively converting its premises. Reduction ad impossibile consists in substituting for the original conclusion its contradictory, and framing a new syllogism in the first figure, having for its premises this contradictory and one of the original premises, and for its conclusion the contradictory of the other original premiss.

As an example of the first mode of reduction, let us take the following syllogism in Cesare:

Ces No circle is a (figure bounded by straight lines);

a Every square is a (figure bounded by straight lines);

re therefore; No square is a circle.

And by the simple conversion of the major premiss, we obtain a syllogism in Celarent with the same conclusion; namely—

Ce No (figure bounded by straight lines) Is a circle;

la Every square is a (figure bounded by straight lines); rent therefore; No square Is a circle.

Again, the following syllogism in Bramantip; namely—
Bram Every triangle is a (plane figure);

an Every (plane figure) is bounded by lines;

tip therefore; Some figures bounded by lines ARE triangles, may be reduced to one in Barbara with a universal conclusion by transposing the premises, thus

Bar Every (plane figure) Is bounded by lines;

ba Every triangle is a (plane figure);

ra therefore; Every triangle is bounded by lines;

from the conclusion of which, by conversion per accidens, we obtain the conclusion of the original syllogism.

As an example of the reductio ad impossibile, we will take the following syllogism in Baroko.

Ba Every triangle is a (plane figure);
rok Some cubes ARE NOT (plane figures);
o therefore; Some cubes ARE NOT triangles.

Then if this conclusion is false, its contradictory must be true; namely

Every cube is a triangle.

And substituting this for the minor premiss, we obtain the following syllogism in Barbara.

Bar Every (triangle) is a plane figure;

ba Every cube Is a (triangle);

ra therefore; Every cube is a plane figure.

But as this conclusion is the contradictory to the minor premiss of the original syllogism, it must be false; and since it has been correctly proved from the premises, one of those premises must be false; but the major premiss is true, therefore it is the minor which is false; and since it is the contradictory to the conclusion of the original syllogism, that conclusion is true. This mode of argument is termed the reductio ad absurdum, and is very frequently employed by Euclid, particularly in the demonstration of propositions from their converse.

By examining the names given to the modes, it will be observed that their initial letters are either B, C, D, or F, and these letters indicate to which of the modes in the first figure any of the modes in one of the other figures is to be reduced. Thus Cesare in the second figure is to be reduced to Celarent in the first, and Bramantip in the fourth to Barbara in the first figure. The letters s or p following a vowel imply that the proposition denoted by it is to be converted, if s, simply, but if p, per accidens; and the letter m signifies that the premises are to be transposed. The letter k after a vowel signifies that the proposition which it denotes is to be omitted, and the contradictory of the conclusion substituted for it, it being in fact the sign of the reductio ad impossibile.

Our limited space will not allow of our entering upon the subject of hypothetical and disjunctive syllogisms, or the very important one of fallacies. We must therefore conclude the foregoing brief sketch of logic with an example of one of the propositions in Euclid, formally demonstrated in syllogisms.

PROPOSITION XXXVI.

HYPOTHESIS.—If parallelograms (ABCD and EFGH) are upon equal bases and between the same parallels,

CONSEQUENCE. They are equal to one another in area.

CONSTRUCTION.-Draw BE and CH.

DEMONSTRATION.

Syllogism 1.

Da (Things which are equal to the same) ARE equal to one an

other. [Ax. 1.]

ri The straight lines BC and EH ARE equal to the same FG. [Hypoth. and I. 34.]

i Therefore; The lines BC and EH ARE equal to one another.

Syllogism 2.

Da (Straight lines which join the adjacent extremities of two equal and parallel straight lines) ARE themselves equal and parallel. [1. 33.]

ri BE and CH ARE (straight lines which join the adjacent extremities of two equal [syl. 1] and parallel straight lines). [hypoth.]

i Therefore; BE and CH ARE themselves equal and parallel.

Syllogism 3.

Da (Parallelograms which are upon the same base and between the same parallels) ARE equal in area. [I. 35.]

ri ABCD and EBCH ARE (parallelograms which are upon the same base and between the same parallels.) [Hypoth. and

syl. 2.]

i Therefore ABCD and EBCH ARE equal in area.

Syllogism 4.

Similar to syl. 3, proving that EFGH and EBCH ARE equal in area.

Syllogism 5.

Da (Things which are equal to the same) ARE equal to one another. [Ax. 1.]

ri ABCD and EFGH ARE (equal in area to the same EBCH.) [Syl. 3 and 4.]

i Therefore ABCD and EFGH ARE equal in area to one another.

The foregoing will sufficiently illustrate the manner in which the propositions of Euclid may be expressed in formal syllogisms, and we should recommend to the student the practice of throwing the more difficult demonstrations into the syllogistic form, as a very useful and beneficial exercise both in logic and geometry.

THE

ELEMENTS OF EUCLID.

BOOK I.

DEFINITIONS.

1. A POINT is that which denotes position, without possessing any magnitude.

2. A LINE is a magnitude, having only one dimension, i.e., length. COROLLARY. The extremities of a line are points, and the intersection of one line with another is also a point.

SCHOLIUM. When a line is cut at any point, the parts of the line between that point and its extremities are termed segments. When the point of section (C) lies between the two extremities (A and B)

of the line, the two portions into which the line is divided (AC and CB) are termed internal segments. But when that point (F) lies in the production of the D line beyond its extremity, the distances from the

E

point (1) to each extremity (FD and FE) are termed external segments. 3. A STRAIGHT LINE is a line which lies evenly (i.e., in the same direction) between its extreme points.

4. A CURVED LINE is a line which continually changes its direction. SCHOLIUM. Whenever the word "line" alone is used throughout this work, it must be taken to mean a straight line.

5. A SURFACE is a magnitude, having only two dimensions, i.e., length and breadth.

COROLLARY. The extremities of a surface are lines, and the intersection of one surface with another is a line.

6. A PLANE SURFACE, or a PLANE, is a surface which lics evenly between its extremities.

SCHOLIUM. The terms point, line, and surface, belong to the class of simple terms, that is to say, they are the names given to simple ideas, by means of which those ideas are conveyed from one mind to another, and consequently (for the reasons stated at length in the introduction) cannot be logically defined. There must always necessarily be a certain amount of difficulty in conveying, for the first time, a simple idea from one mind to another with perfect accuracy. But the idea having been once accurately conveyed, and associated with a certain name, can at any future time be readily recalled to our mind by the mention of that name alone. The firs:

B

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