A Corollary is an inference made immediately from a proposition. A Scholium is a note or explanatory observation. A Lemma is a preparatory proposition borrowed from another part of the same subject, and introduced for the purpose of establishing a more important proposition. The Converse of a proposition is when the hypothesis of a former proposition becomes the conclusion, or predicate, of the latter proposition. The Contrary of a proposition is when that which the proposition assumes, is denied. Direct Demonstration is when the very thing asserted is proved to be true. Indirect Demonstration is when all other cases, or conditions, except the one in question, are proved not to be true, and the inference is made-therefore the very thing in question must be true; the assumption being that one out of several, or many, must be right. The Position only of a line is meant, when the line is said to be given. The Length only of a line is meant, when the line is said to be finite. The Base of a figure is the side on which it appears to stand, but each side, in turn, with the position of the figure changed, may become the base. The Vertex is the highest angular point of a figure: with a change of position in the figure, each angle may be named the vertical angle. The Subtend of an angle is the side stretching across opposite to the angle. The Hypotenuse is the subtend to a right angle. The Perpendicular is the line forming with the base a right angle lines are perpendicular to each other when at the point of junction they form a right angle. A Figure is applied to a straight line when the line forms one of its boundaries. The Altitude of a figure is the perpendicular distance from the side or angle opposite to the base, to the base itself, or to the base produced. A Diagonal is a line joining two opposite angular points. The Complement of an angle is what is wanted to make an acute angle equal to a right angle, or to 90°. The Supplement of an angle is what is wanted to make an angle equal to two right angles, or to 180°. The Explement of an angle is what is wanted to make an angle equal to four right angles, or to 360°. The Complements of a Parallelogram, when the parallelogram is bisected by its diagonal, and subsidiary parallelograms are formed by two lines, one, parallel to one side, and the other, parallel to the other side, and both intersecting the diagonal,the complements of the parallelogram are those subsidiary parallelograms through which the diagonal does not pass; and these, with the subsidiary parallelograms through which the diagonal does pass, fill up or complete the whole parallelogram. The Area of a Figure is the quantity of surface contained in it, reckoned in square units, as square inches, square feet, &c. A locus in Plane Geometry is a straight line, or a plane curve, every point of which, and none else, satisfies a certain condition. SECTION IV. NATURE OF GEOMETRICAL REASONING. THE Demonstrations in Euclid's Elements of Geometry consist of arguments or reasonings by which the assertions made in the propositions are proved to be true. Thus, in the 15th Prop., bk. i., the assertion is made that "the opposite, or vertical angles, formed by two intersecting lines, are equal;" and the demonstration shows by argument, or reasoning, founded upon truths already admitted or proved, that the assertion itself must be received as true. When fully stated, each argument contains both the thing which is proved, and the means by which the proof is established: and as in the arrangement of the parts of an argument the means of proof usually precede the thing proved, they are named the premisses; and the thing proved is named the conclusion or inference. Thus, in Prop. 1, bk. i., the premisses are-1st, things equal to the same thing are equal to each other; 2nd, the line A C, and also the line B.C, are each equal to the same line A B; and 3rd, the inference, conclusion, or thing proved, is, that the line AC equals the line BC, or, adhering more strictly to the forms of reasoning, the two lines A C and B C are equal to each other. Here in the premisses two things are laid down, or granted to be true: as,-"things equal to the same thing are equal to each other," this is one truth; "the line AC equals the line A B, and the line B C also equals the same line A B,"-this is another truth; and from the two things thus declared to be true, there is made in the conclusion the necessary and unavoidable inference, therefore AC equals BC, i. e., the two lines are equal to each other. 66 The subject of the conclusion, "the two lines AC and BC," is called the minor term; the predicate of the conclusion, equal to each other," is called the major term. are There are three terms in the premisses, the major and the minor terms, and a third term with which as with a standard the major and the minor terms are compared. This third term, being the medium of the comparison, is named the middle term: it enters into the major premiss as the subject, and into the minor premiss as the predicate. In the example given, "things equal to the same," is the middle term, being the subject of the major premiss-"things equal to the same are equal to each other," and the predicate of the minor premiss-"the lines AC and BC are each equal to the same line A B." The premiss in which the major term-i. e., the predicate of the conclusion-appears, is called the major premiss; that in which the minor term, or subject of the conclusion, appears, is the minor premiss. Thus in the argument already given: Major premiss, Because things equal to the same, are equal to each other; Minor premiss, and because the two lines AC and BC, are each equal to the same line A B; Conclusion, therefore the two lines A C and BC are equal to each other. Here, "the two lines AC, BC" is the subject, and "equal to each other" the predicate of the conclusion; 66 The two lines AC, BC" is the subject, and each equal to the same line AB," the predicate of the minor premiss; Things equal to the same line AB" the subject, and "equal to each other" the predicate of the major premiss. Thus, "equal to each other" is the major term; "the two lines AC, BC" the minor term; and "things equal to the same line AB" the middle term of the argument. In this mode of reasoning, it is seen that assertions are broadly made; and we may ask, on what evidence are these assertions themselves to be received as true? The first kind of evidence is from the definition of the thing; thus, we define a triangle to be a figure bounded by three sides; and if, of any figure placed before us, we can affirm that it has three sides exactly, the conclusion is inevitable, that this figure also is a triangle. The second kind of evidence is from the axioms, or truths so plain that they need no proof: for example, we receive as undeniable, that, if equals be added to equals, the wholes are equal; and we argue, if to the line A D, or to its equal the line BC, we add another line E F, then the whole line made up of AD EF, will equal the whole line made up of BC + E F. The third kind of evidence is from the hypothesis, or supposition, which we make as the condition of our assertion: we declare, "in an isosceles triangle, the angles at the base are equal;" the very words, though not in the exact form of an hypothesis, directly imply the supposition, "if a triangle is isosceles," "then the angles at the base are equal.' An isosceles triangle is here taken as the starting point of the reasoning;-and though, for the demonstration of the inference, "the angles at the base are equal," it is necessary to draw various lines which are not mentioned in the hypothesis, the conclusion at which we arrive is altogether dependent on the hypothesis. The fourth kind of evidence is from proof already given; for what has once been established, may afterwards be taken for granted. For instance, when we have once established the truth, that "the interior angles of every triangle are together equal to two right angles," and we afterwards come to a proposition in the demonstration of which we need this established truth, we do not again go through all the steps by which the equality of the sum of the interior angles of a triangle to two right angles has been proved, but, without going down again to the bottom of the ladder before we make a step higher, we start from the step we had already gained, and at once take up our position on a more advanced truth. But the Principle of Geometrical Reasoning is, that from two propositions established or received as true, a third proposition, or inference, shall be made. Now, that this may be done, there must be something in common contained in both the propositions, with which common thing, the other two things are compared: we say All the triangle is in the circle, All the square is in the triangle, therefore, All the square is in the circle: the common term of comparison here is "the triangle," and our inference is correct. But if we say All the triangle is in the circle, All the square is in the circle; and infer, All the square is in the triangle; this may be, or may not be,—for it may happen that only a part of the square is in the triangle. The fault of the apparent argument is, there is no proper term of comparison,-no middle term which is at the same time the subject of the major premiss, and the predicate of the minor premiss. To have the argument sound, we say— Major P. All the triangle is in the circle, Minor P. All the square is in the triangle, When a connexion is thus declared to exist between the premisses and the conclusion,—that is, when reasons are stated and an inference made,-this mode of argument receives the name of a Syllogism; for a Syllogism is a bringing together into one view the two steps of the reasoning on which a truth depends, and the truth itself; or, as Whately, in his Elements of Logic, p. 52, defines a Syllogism, it is "an argument so expressed, that the conclusiveness of it is manifest from the mere force of the expression, i. e., without considering the meaning of the terms: e. g., in this Syllogism "Every Y is X, Z is Y, therefore Z is X; "the Conclusion is inevitable, whatever terms X, Y, and Z |