tains 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 selfevident 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 enthymeme, 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. I. 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 shewn 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 AB is equal to the straight 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 (σnuɛĩov) 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. The essential parts of every Problem are the data and quæsita; and of a theorem, the hypothesis and predicate. These should be accurately discriminated by the student in the first place in order to avoid confusion of thought. It may also assist him in his progress, to observe that every complete proposition may be divided into six parts, as Proclus has pointed out in his Commentary on the First Book of Euclid's Elements. 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 property of the circle is employed in these books except 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. III. 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 A B 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 & theorem which results from the demonstration of a proposition. Prop. vi. is the converse of one part of Prop. v. One proposition is defined to be the converse of another when the hypothesis of the former becomes the predicate of the latter; and vice versa. There is besides this, another kind of conversion, when a theorem has several hypotheses and one predicate; by assuming the predicate and one, or more than one of the hypotheses, some one of the hypotheses may be inferred as the predicate of the converse. In this manner, Prop. VIII. is the converse of Prop. IV. It may here be observed, that converse theorems are not universally true: as for instance, the following direct proposition is universally true; If two triangles have their three sides respectively equal, the three angles of each shall be respectively equal." But the converse is not universally true; namely, "If two triangles have the three angles in each respectively equal, the three sides are respectively equal." Converse theorems require, in some instances, the consideration of other conditions than those which enter into the proof of the direct theorem. Converse and contrary propositions are by no means to be confounded; the contrary proposition denies what is asserted, or asserts what is denied, in the direct proposition, but the subject and predicate in each are the same. A contrary proposition is a completely contradictory proposition, and the distinction consists in this-that two contrary propositions may both be false, but of two contradictory propositions, one of them must be true, and the other false. It may here be remarked, that one of the most common intellectual mistakes of learners, is to imagine that the denial of a proposition, is a legitimate ground for affirming the contrary as true: whereas the rules of sound reasoning allow, that the affirmation of a proposition as true, only affords a ground for the denial of the contrary as false. Prop. VI. is the first instance of indirect demonstrations, and they are more suited for the proof of converse propositions. All those propositions which are demonstrated ex absurdo, are properly analytical demonstrations, according to the Greek notion of analysis, which first supposed the thing required, to be done, or to be true, and then shewed the consistency or inconsistency of this construction or hypothesis with truths admitted or already demonstrated. In indirect demonstrations, where hypotheses are made which are not true and contrary to the truth stated in the proposition, it seems desirable that a form of expression should be employed different from that in which the hypotheses are true. In all cases therefore, whether noted by Euclid or not, the words if possible have been introduced, or some such qualifying expression, as in Euc. 1. 6, so as not to leave upon the mind of the learner the impression, that the hypothesis which contradicts the proposition, is really true. An argument which ends in an absurdity, is that in which, by proving the contradictory of the assertion contained in the predicate, to be evidently false, the assertion in the predicate itself is true. The logical principles on which it depends are these: from the contradictory of the assertion we wish to prove, and from another assertion, the truth of which is manifest, we deduce a conclusion evidently false. As the conclusion is false, one of the premisses must be false; for a true conclusion cannot follow from a false premiss: but one of the premisses is supposed to be evidently true: therefore the false premiss must be the contradictory of the assertion, the truth of which we wish to establish; and as the contradictory of the assertion is false, the assertion itself is true. 1 In general, with respect to indirect demonstrations, it may be questioned whether they ought ever to be admitted as a legitimate mode of proof in a primary and fundamental proposition. Indirect demonstrations are properly and most effectually applied in proving the converse of a proposition, which has been demonstrated by a direct appeal to assumed or demonstrated principles. To make a negative property or an indirect demonstration the basis of a positive doctrine, seems to be an inversion of the natural process the mind pursues in the investigation of truth, and to leave the doctrine exposed to the objections which may be made from the illogical attempt, to prove the affirmative from a negative. Prop. VII. The enunciation in the text was altered into that form by Simson. Euclid's is, Ἐπὶ τῆς αὐτῆς εὐθείας, δυσὶ ταῖς αὐταῖς εὐθείαις ἄλλαι δύο εὐθεῖαι ἴσαι ἑκατέρα ἑκατέρᾳ οὐ συσταθήσονται πρὸς ἄλλῳ καὶ ἄλλῳ σημείῳ ἐπὶ τὰ αὐτὰ μέρη, τὰ αὐτὰ πέρατα ἔχουσαι ταῖς ἐξ ἀρχῆς εὐθείαις. Prop. VIII. When the three sides of one triangle are shewn to coincide with the three sides of any other, the equality of the triangles is at once obvious. This however, is not stated at the conclusion of Prop. vIII. or of Prop. xxvI. For the equality of the areas of two coincident triangles, reference is always made by Euclid to Prop. IV. A direct demonstration may be given of this proposition, and Prop. vii. may be dispensed with altogether. Let the triangles ABC, DEF be so placed that the base BC may coincide with the base EF, and the vertices A, D may be on opposite sides of EF. Join AD. Then because EAD is an isosceles triangle, the angle EAD is equal to the angle EDA; and because CDA is an isosceles triangle, the angle CAD is equal to the angle CDA. Hence the angle EAF is equal to the angle EDF, (ax. 2 or 3): or the angle BDC is equal to the angle EDF. Prop. ix. If BA, AC be in the same straight line This problem then becomes the same as Prob. xI., which may be regarded as drawing a line which bisects an angle equal to two right angles. If FA be produced in the fig.. Prop. x, it bisects the angle which is the defect of the angle BAC from four right angles. By means of this problem, any angle may be divided into four, eight, sixteen, &c. equal angles.. · Prop. x. A finite straight line may, by this problem, be divided into four, eight, sixteen, &c. equal parts. Prop. XI. When the point is at the extremity of the line; by the second postulate the line may be produced, and then the construction applies. See note on Euc. III. 31. The distance between two points, is the straight line which joins the points; but the distance between a point and a straight line,, is. the shortest line which can be drawn from the point to the line. From this Prop. it follows that only one perpendicular can be drawn, from a given point to a given line; and this perpendicular may be shewn to be less than any other line which can be drawn from the given point to the given line; and of the rest, the line which is nearer to the perpendicular is less than one more remote from it also only two equal straight lines can be drawn from the same point to the line, one on each side of the perpendicular or the least. This property is analogous to Euc. 11. 7, 8. The corollary to this proposition is not in the Greek text, but was added by Simson, namely:-"That two straight lines cannot have a common segment." It seems to have been assumed in the demonstration of Euc. 1. 4. Prop. XII. The third postulate requires that the line CD should be drawn before the circle can be described with the center C, and radius CD. Prop. XIV. is the converse of Prop. XII. Upon the opposite sides of it." If these words were omitted, it is possible for two lines to make with a third, two angles, which together are equal to two right angles, in such a manner that the two lines shall not be in the same straight line. The line BE may be supposed to fall above, as in Euclid's figure, or below the line BD, and the demonstration is the same in form. Prop. xv. is the development of the definition of an angle. If the lines at the angular point be produced, the produced lines have the same inclination to one another as the original lines, but in a different position. The converse of this Proposition is not proved by Euclid, namely:-If the vertical angles made by four straight lines at a point be respectively equal to each other, each pair of opposite lines shall be in the same straight line. Prop. XVII. appears to be only a corollary to the preceding proposition, and to be introduced to explain Axiom XII., of which it is the converse. The exact truth respecting the angles of a triangle is proved in Prop. XXXII. Prop. xvIII. It may here be remarked, for the purpose of guarding the student against a very common mistake, that in this proposition and in the converse of it, the hypothesis is stated before the predicate. Prop. xIx. is the converse of Prop. xvII. Prop. xix. bears the same relation to Prop. XVIII., as Prop. vi. does to Prop. v. Prop. xx. The following corollary arises from this proposition : A straight line is the shortest distance between two points. For the straight line BC is always less than BA and AC, however near the point A may be to the line BC. And also the sum of the three sides taken together, is greater than the double of any one side, but less than the double of any two sides.. It may be easily shewn from this proposition,, that the difference of any two sides of a triangle is less than the third side. |