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This axiom is the criterion of Geometrical equality, and is essentially different from the criterion of Arithmetical equality. Two geometrical magnitudes are equal, when they coincide or may be made to coincide : two abstract numbers are equal, when they contain the same aggregate of units; and two concrete numbers are equal, when they contain the same number of units of the same kind of magnitude. It is at once obvious, that Arithmetical representations of Geometrical magnitudes are not admissible in Euclid's criterion of Geometrical Equality, as he has not fixed the unit of magnitude of either the straight line, the angle, or the superficies. Perhaps Euclid intended that the first seven axioms should be applicable to numbers as well as to Geometrical magnitudes, and this is in accordance with the words of Proclus, who calls the axioms, common notions, not peculiar to the subject of Geometry.
Several of the axioms may be generally exemplified thus:
Axiom 1. If the straight line AB be equal Á to the straight line CD; and if the straight
-CD line EF be also equal to the straight line CD;
E F then the straight line AB is equal to the straight line EF.
Axiom it. If the line AB be equal to the line A__BC__ CD; and if the line EF be also equal to the line GH: then the sum of the lines AB and EF E F G H is equal to the sum of the lines CD and GH. Axiom 111. If the line AB be equal to the
A B C D line CD; and if the line EF be also equal to the line GH; then the difference of AB and EF, E_F G _ 1 is equal to the difference of CD and GH.
Axiom iv. admits of being exemplified under the two following forms :
1. If the line AB be equal to the line CD; A B C D and if the line EF be greater than the line GH; then the sum of the lines AB and EF is greater E F G H than the sum of the lines CD and GH.
2. If the line AB be equal to the line CD; A B C D and if the line EF be less than the line GH; -then the sum of the lines AB and EF is less E F G H than the sum of the lines CD and GH.
Axiom v. also admits of two forms of exemplification.
1. If the line AB be equal to the line CD; A B C D and if the line EF be greater than the line GH; then the difference of the lines AB and EF is E F G H greater than the difference of CD and GH.
2. If the line AB be equal to the line CD; A B c__D and if the line EF be less than the line GH; then the difference of the lines AB and EF is E_ FG___ H less than the difference of the lines CD and GH.
The axiom, “If unequals be taken from equals, the remainders are unequal,” may be exemplified in the same manner.
Axiom vi. If the line AB be double of the A_ line CD; and if the line EF be also double of
C D the line CD;
E_F then the line AB is equal to the line EF.
Axiom vII. If the line AB be the half of A B the line CD; and if the line EF be also the
C_ D half of the line CD;
E F then the line AB is equal to the line EF.
It may be observed that when equal magnitudes are taken from unequal magnitudes, the greater remainder exceeds the less remainder by as much as the greater of the unequal magnitudes exceeds the less.
If unequals be taken from unequals, the remainders are not always unequal; they may be equal : also if unequals be added to unequals the wholes are not always unequal, they may also be equal.
Axiom ix. The whole is greater than its part, and conversely, the part is less than the whole. This axiom appears to assert the contrary of the eighth axiom, namely, that two magnitudes, of which one is greater than the other, cannot be made to coincide with one another.
Axiom 8. The property of straight lines expressed by the tenth axiom, namely, “that two straight lines cannot enclose a space,” is obviously implied in the definition of straight lines; for if they enclosed a space, they could not coincide between their extreme points, when the two lines are equal.
Axiom xi. This axiom has been asserted to be a demonstrable theorem. As an angle is a species of magnitude, this axiom is only a particular application of the eighth axiom to right angles. Axiom XII. See the notes on Prop. xxix. Book 1.
ON THE PROPOSITIONS. WHENEVER a judgment is formally expressed, there must be something respecting which the judgment is expressed, and something else which constitutes the judgment. The former is called the subject of the proposition, and the latter, the predicate, which may be anything which can be affirmed or denied respecting the subject.
The propositions in Euclid's Elements of Geometry may be divided into two classes, problems and theorems. A proposition, as the term imports, is something proposed; it is a problem, when some Geometrical construction is required to be effected: and it is a theorem when some Geo. metrical property is to be demonstrated. Every proposition is naturally divided into two parts; a problem consists of the data, or things given; and the quæsita, or things required: a theorem, consists of the subject or hypothesis, and the conclusion, or predicate. Hence the distinction between a problem and a theorem is this, that a problem consists of the data and the quæsita, and requires solution: and a theorem consists of the hypothesis and the predicate, and requires demonstration.
All propositions are affirmative or negative; that is, they either assert some property, as Euc. 1. 4, or deny the existence of some property, as Euc. i. 7; and every proposition which is affirmatively stated has a contradictory corresponding proposition. If the affirmative be proved to be true, the contradictory is false.
All propositions may be viewed as (1) universally affirmative, or universally negative ; (2) as particularly affirmative, or particularly negative.
The connected course of reasoning by which any Geometrical truth is established is called a demonstration. It is called a direct demonstration when the predicate of the proposition is inferred directly from the premisses, as the conclusion of a series of successive deductions. The demonstration is called indirect, when the conclusion shows that the introduction of any other supposition contrary to the hypothesis stated in the proposition, necessarily leads to an absurdity.
It has been remarked by Pascal, that “Geometry is almost the only subject as to which we find truths wherein all men agree; and one cause of this is, that Geometers alone regard the true laws of demonstration."
These are enumerated by him as eight in number. “1. To define nothing which cannot be expressed in clearer terms than those in which it is already expressed. 2. To leave no obscure or equivocal terms undefined. 3. To employ in the definition no terms not already known. 4. To omit nothing in the principles from which we argue, unless we are sure it is granted. 5. To lay down no axiom which is not perfectly evident. 6. To demonstrate nothing which is as clear already as we can make it. 7. To prove every thing in the least doubtful by means of self-evident axioms, or of propositions already demonstrated. 8. To substitute mentally the definition instead of the thing defined.” Of these rules, he says, “the first, fourth and sixth are not absolutely necessary to avoid error, but the other five are indispensable; and though they may be found in books of logic, none but the Geometers have paid any regard to them.”
The course pursued in the demonstrations of the propositions in Euclid's Elements of Geometry, is always to refer directly to some expressed principle, to leave nothing to be inferred from vague expressions, and to make every step of the demonstrations the object of the understanding.
It has been maintained by some philosophers, that a genuine definition contains some property or properties which can form a basis for demonstration, and that the science of Geometry is deduced from the definitions, and that on them alone the demonstrations depend. Others have maintained that a definition explains only the meaning of a term, and does not embrace the nature and properties of the thing defined.
If the propositions usually called postulates and axioms are either tacitly assumed or expressly stated in the definitions; in this view, demonstrations may be said to be legitimately founded on definitions. If, on the other hand, a definition is simply an explanation of the meaning of a term, whether abstract or concrete, by such marks as may prevent a misconception of the thing defined ; it will be at once obvious that some constructive and theoretic principles must be assumed, besides the definitions to form the ground of legitimate demonstration. These principles we conceive to be the postulates and axioms. The postulates describe constructions which may be admitted as possible by direct appeal to our experience; and the axioms assert general theoretic truths so simple and self-evident as to require no proof, but to be admitted as the assumed first principles of demonstration. Under this view all Geometrical reasonings proceed upon the admission of the hypotheses assumed in. the definitions, and the unquestioned possibility of the postulates, and the truth of the axioms.
Deductive reasoning is generally delivered in the form of an enthymenie, or an argument wherein one enunciation is not expressed, but is readily supplied by the reader: and it may be observed, that although this is the ordinary mode of speaking and writing, it is not in the strictly syllogistic form; as either the major or the minor premiss only is formally stated before the conclusion: Thus in Euc. 1. 1.
Because the point A is the center of the circle BCD; therefore the straight line AB is equal to the straight line AC. The premiss here omitted, is: all straight lines drawn from the center of a circle to the circumference are equal.
In a similar way may be supplied the reserved premiss in every enthymeme. The conclusion of two enthymemes may form the major and minor premiss of a third syllogism, and so on, and thus any process of reasoning is reduced to the strictly syllogistic form. And in this way it is shewy
that the general theorems of Geometry are demonstrated by means of syllogisms founded on the axioms and definitions.
Every syllogism consists of three propositions, of which, two are called the premisses, and the third, the conclusion. These propositions contain three terms, the subject and predicate of the conclusion, and the middle term which connects the predicate and the conclusion together. The subject of the conclusion is called the minor, and the predicate of the conclusion is called the major term, of the syllogism. The major term appears in one premiss, and the minor term in the other, with the middle term, which is in both premisses. That premiss which contains the middle term and the major term, is called the major premiss; and that which contains the middle term and the minor term, is called the minor premiss of the syllogism. As an example, we may take the syllogism in the demonstration of Prop. 1, Book 1, wherein it will be seen that the middle term is the subject of the major premiss and the predicate of the minor. Major premiss: because the straight line A B is equal to thestraight line AC; Minor premiss: and, because the straight line BC is equal to the straight line AB; Conclusion: therefore the straight line BC is equal to the straight line 4C. Here, BC is the subject, and AC the predicate of the conclusion.
BC is the subject, and AB the predicate of the minor premiss.
AB is the subject, and AC the predicate of the major premiss. Also, AC is the major term, BC the minor term, and AB the middle term of the syllogism.
In this syllogism, it may be remarked that the definition of a straight line is assumed, and the definition of the Geometrical equality of two straight lines ; also that a general theoretic truth, or axiom, forms the ground of the conclusion. And further, though it be impossible to make any point, mark or sign (onuecov) which has not both length and breadth, and any line which has not both length and breadth; the demonstrations in Geometry do not on this account become invalid. For they are pursued on the hypothesis that the point has no parts, but position only : and the line has length only, but no breadth or thickness : also that the surface has length and breadth only, but no thickness: and all the conclusions at which we arrive are independent of every other consideration.
The truth of the conclusion in the syllogism depends upon the truth of the premisses. If the premisses, or only one of them be not true, the conclusion is false. The conclusion is said to follow from the premisses; whereas, in truth, it is contained in the premisses. The expression must be understood of the mind apprehending in succession, the truth of the premisses, and subsequent to that, the truth of the conclusion ; so that the conclusion follows from the premisses in order of time as far as reference is made to the mind's apprehension of the whole argument.
Every proposition, when complete, may be divided into six parts, as Proclus has pointed out in his commentary.
1. The proposition, or general enunciation, which states in general terms the conditions of the problem or theorem.
2. The exposition, or particular enunciation, which exhibits the subject of the proposition in particular terms as a fact, and refers it to some diagram described.
3. The determination contains the predicate in particular terms, as it is pointed out in the diagram, and directs attention to the demonstration, by pronouncing the thing sought.
4. The construction applies the postulates to prepare the diagram for the demonstration.
5. The demonstration is the connexion of syllogisms, which prove the truth or falsehood of the theorem, the possibility or impossibility of the problem, in that particular case exhibited in the diagram.
6. The conclusion is merely the repetition of the general enunciation, wherein the predicate is asserted as a demonstrated truth.
Prop. 1. In the first two Books, the circle is employed as a mechanical instrument, in the same manner as the straight line, and the use made of it rests entirely on the third postulate. No properties of the circle are discussed in these books beyond the definition and the third postulate. When two circles are described, one of which has its center in the circumference of the other, the two circles being each of them partly within and partly without the other, their circumferences must intersect each other in two points; and it is obvious from the two circles cutting each other, in two points, one on each side of the given line, that two equilateral triangles may be formed on the given line.
Prop. II. When the given point is neither in the line, nor in the line produced, this problem admits of eight different lines being drawn from the given point in different directions, every one of which is a solution of the problem. For, 1. The given line has two extremities, to each of which a line may be drawn from the given point. 2. The equilateral triangle may be described on either side of this line. 3. And the side BD of the equilateral triangle ABD may be produced either way.
But when the given point lies either in the line or in the line produced, the distinction which arises from joining the two ends of the line with the given point, no longer exists, and there are only four cases of the problem.
The construction of this problem assumes a neater form, by first describing the circle CGH with center B and radius BC, and producing DB the side of the equilateral triangle DBA to meet the circumference in G: next, with center D and radius DG, describing the circle GKL, and then producing DA to meet the circumference in L.
By a similar construction the less of two given straight lines may be produced, so that the less together with the part produced may be equal to the greater.
Prop. 111. This problem admits of two solutions, and it is left undetermined from which end of the greater line the part is to be cut off.
By means of this problem, a straight line may be found equal to the sum or the difference of two given lines.
Prop. iv. This forms the first case of equal triangles, two other cases are proved in Prop. viii. and Prop. XXVI.
The term base is obviously taken from the idea of a building, and the same may be said of the term altitude. In Geometry, however, these terms are not restricted to one particular position of a figure, as in the case of a building, but may be in any position whatever.
Prop. v. Proclus has given, in his commentary, a proof for the equality of the angles at the base, without producing the equal sides. The construction follows the same order, taking in AB one side of the isosceles triangle ABC, a point D and cutting off from AC a part AE equal to AD, and then joining CD and BE.
A corollary is a theorem which results from the demonstration of a proposition. * Prop. vi. is the converse of one part of Prop. v. One proposition