ON THE PROPOSITIONS. THE deductive truths of Geometry are called propositions, which are 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 Geometrical 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 premisses, hypothesis, or the properties admitted; and the conclusion, or predicate, or properties to be demonstrated. Hence the distinction between a problem and a theorem is this, that a problem consists of data and quæsita, and requires solution: and a theorem consists of the hypothesis and the predicate, and requires demonstration. 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 shews that the introduction of any other supposition contrary to the hypothesis stated in the proposition, necessarily leads to an absurdity. 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 grounds 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. The general theorems of Geometry are demonstrated by means of syllogisms founded on the axioms and definitions. The form of syllogism employed in Geometrical reasonings is of the simplest character. 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 conclu. sion 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 first 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; 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, (σnμeîov) 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. 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. 4. The construction applies the postulates to prepare the diagram for the de monstration. 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 Books I and II, 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 or even alluded to in these books beyond the definition and the third postulate. One circle may fall within or without another entirely, or the circumferences may intersect each other, as when the centre of one circle is in the circumference of the other; 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. 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. Prop. IV. This forms the first case of equal triangles, two other cases are proved in Props. VIII and XXVI. A distinction ought to be made between equal triangles and equivalent triangles, the former including those whose sides and angles mutually coincide, the latter those whose areas only are equivalent. 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 a point D and cutting off from AC a part AE equal to AB, and then joining CD and BE. A corollary is a theorem which results from the demonstration of a proposition, and generally is so obvious as to require no formal proof. 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 assumed in the direct proposition, but the subject and predicate in each are the same. 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. 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. Prop. IX. By means of this problem, any angle may be divided into four, eight, sixteen, &c. equal angles. Prop. x. Any 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. Prop. XIII. It is manifest that the lines which bisect the angles ABC and ABD are at right angles to each other. Prop. XIV is the converse of Prop. XIII. "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. Prop. xv is the developement 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. Prop. XVI. 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. The exact Prop. xvii appears to be only a corollary to the preceding proposition, and it seems to be introduced to explain Axiom XII, of which it is the converse. truth respecting the angles of a triangle is proved in Prop. XXXII. Prop. XIX is the converse of Prop. XVIII. It may be remarked, that Prop. XIX bears the same relation to Prop. XVIII, as Prop. vi does to Prop. v. Prop. xx-xxI. "Proclus, in his commentary, relates, that the Epicureans derided this proposition, as being manifest even to asses, and needing no demonstration; and his answer is, that though the truth of it be manifest to our senses, yet it is science which must give the reason why two sides of a triangle are greater than the third. But the right answer to this objection against this and Prop. XXI, and some other plain propositions, is, that the number of axioms ought not to be increased without necessity, as it must be if these propositions be not demonstrated. Mons. Clairault, in the preface to his Elements of Geometry, published in French at Paris, 1741, says, 'that Euclid has been at the pains to prove, that the two sides of a triangle which is included within another, are together less than the two sides of the triangle which includes it.' But he has forgot to add this condition, viz. that the triangles must be upon the same base: because, unless this be added, the sides of the included triangle may be greater than the sides of the triangle which includes it, in any ratio which is less than that of two to one, as Pappus Alexandrinus has demonstrated in Prop. 3, Book 111 of his Mathematical Collections." Simson. Prop. XXII. When the sum of two of the lines is equal to, and when it is less than, the third line; let the diagrams be described, and they will exhibit the impossibility implied by the restriction laid down in the proposition. Prop. XXIII. CD might be taken equal to CE and the construction effected by means of an isosceles triangle. It would, however, be less general than Euclid's. Prop. XXIV. Simson makes the angle EDG at D in the line ED, the side which is not the greater of the two ED, DF; otherwise, three different cases would arise, as may be seen by forming the different figures. The point G might fall below or upon the base EF produced as well as above it. Prop. XXIV and Prop. xxv bear to each other the same relation as Prop. IV and Prop. VIII. Prop. XXVI. This forms the third case of the equality of two triangles. Every triangle has three sides and three angles, and when any three of one triangle are given equal to any three of another, the triangles may be proved to be equal to one another, whenever the three magnitudes given in the hypothesis are independent of one another. Prop. IV contains the first case, when the hypothesis consists of two sides and the included angle of each triangle. Prop. VIII contains the second, when the hypothesis consists of the three sides of each triangle. Prop. XXVI contains the third, when the hypothesis consists of two angles, and one side either adjacent to the equal angles, or opposite to one of the equal angles in each triangle. There is another case, not proved by Euclid, when the hypothesis consists of two sides and one angle in each triangle, but these not the angles included by the two given sides in each triangle. This case however is only true under a certain restriction. Prop. XXVII. Alternate angles are defined to be the two angles which two straight lines make with another at its extremities, but upon opposite sides of it. Prop. XXVIII. One angle is called "the exterior angle," and another "the interior and opposite angle," when they are formed on the same side of a straight line which falls upon or intersects two other straight lines. It is also obvious that on each side of the line, there will be two exterior and two interior and opposite angles. The exterior angle EGB has the angle GHD for its corresponding interior and opposite angle: also the exterior angle FHD has the angle HGB for its interior and opposite angle. Prop. XXIX is the converse of Prop. XXVII and Prop. XXVIII. As the definition of parallel straight lines simply describes them by a statement of the negative property, that they never meet; it is necessary that some positive property of parallel lines should be assumed as an axiom, on which reasonings on such lines may be founded. Euclid has assumed the statement in the twelfth axiom, which has been objected to, as not being self-evident. A stronger objection appears to be, that the converse of it forms Prop. 17, Book 1; for both the assumed axiom and its converse, should be so obvious as not to require formal demonstration. Simson has attempted to overcome the objection, not by any improved definition and axiom respecting parallel lines; but, by considering Euclid's twelfth axiom to be a theorem, and for its proof, assuming two definitions and one axiom, and then demonstrating five subsidiary Propositions. Instead of Euclid's twelfth axiom, the following has been proposed as a more simple property for the foundation of reasonings on parallel lines; namely, "If a straight line fall on two parallel straight lines, the alternate angles are equal to one another." In whatever this may exceed Euclid's definition in simplicity, it is liable to a similar objection, being the converse of Prop. 27, Book 1. Professor Playfair has adopted in his Elements of Geometry, that "Two straight lines which intersect one another cannot be both parallel to the same straight line." This apparently more simple axiom follows as a direct inference from Prop. 30, Book 1. But one of the least objectionable of all the definitions which have been proposed on this subject, appears to be that which simply expresses the conception of equidistance. It may be formally stated thus: "Parallel lines are such as lie in the same plane, and which neither recede from, nor approach to, each other." This includes the conception stated by Euclid, that parallel lines never meet. Dr Wallis observes on this subject, "Parallelismus et æquidistantia vel idem sunt, vel certe se mutuo comitantur." As an additional reason for this definition being preferred, it may be remarked that the meaning of the terms ypaμμai wapádλndo, suggests the exact idea of such lines. Axiom X1 and XII, in some manuscripts of Euclid, are found placed respectively as the fourth and the fifth postulate. An account of thirty methods which have been proposed at different times for avoiding the difficulty in the twelfth axiom, will be found in the appendix to Mr Thompson's "Geometry without Axioms." |