Prop. xxI. It must be remarked that the two lines drawn to a point within the triangle must be drawn from the extremities of the base; otherwise, it is possible for two lines to be drawn from two points in the base, such that their sum shall be greater than the two sides of the triangle. This however is not possible when the given triangle is equilateral, or isosceles having the base less than either of the equal sides. In other cases it is always possible to find a point within a triangle such that two lines may be drawn from points in the base, having their sum greater than the two sides of the triangle. See Pappi. Math. Coll. Lib. 111. Props. 28-34. The demonstration of this proposition is an instance of what is called the argument à fortiori, because the premisses are more than sufficient to prove the conclusion. The argument in Euclid, namely-Because BA, AC are greater than BE, EC; and because BE, EC are greater than BD, DC; therefore BA, AC are greater than BD, DC; does not constitute a legitimate syllogism, because there is no middle term, and it involves more than a syllogism can express. The following additional proposition must be added before the form can be made completely logical: if BE, EC be a magnitude greater than BD, DC, then every magnitude greater than BE, EC, is also greater than BD, DC. The legitimate logical form then becomes. Because BA, AC contain BE, EC and more, and because BE, EC contain BÐ, DC and more, therefore BA, AE contain, BD, DC and more, and consequently BA, AC are greater than BD, DC. 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. The same remark may be made here, as was made under the first Proposition, namely:-If one circle lie partly within and partly without another circle, the circumferences of the circles intersect each other in two points. 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, but is more convenient in practice. Prop. xxiv. Simson makes the angle EDG at D in the line ED, the side which 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. xxXVI, 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, thus: If two triangles have two sides of one of them, equal to two sides of the other, each to each, and have also the angles opposite to one of the equal sides in each triangle, equal 1 to one another, and if the angles opposite to the other equal sides be both acute, or both obtuse angles; then shall the third sides be equal in each triangle, as also the remaining angles of the one to the remaining angles of the other. Let ABC, DEF be two triangles which have the sides. AB, AC equal to the two sides DE, DF, each to each, and the angle ABC equal to the angle DEF; then, if the angles ACB, DEF, be both acute, or both obtuse angles, the third side BC shall be equal to the third side EF, and also the angle BCA to the angle EFD, and the angle BAC to the angle EDF. First. Let the angles ACB, DFE opposite to the equal sides AB, DE, be both acute angles.. If BC be not equal to EF, let BC be the greater, and from BC, cut off BG equal to EF, and join AG. Then in the triangles ABG, DEF, Euc. 1. 4, AG is equal to DF, and the angle AGB to DFE. But since AC is equal to DF, AG is equal to AC: and therefore the angle ACG is equal to the angle AGC, which is also an acute angle. But because AGC, AGB are together equal to two right angles, and that AGC is an acute angle, AGB must be an obtuse angle; which is absurd. Wherefore, BC is not unequal to EF, that is, BC is equal to EF, and also the remaining angles of one triangle to the remaining angles of the other. Secondly. Let the angles ACB; DFE, be both obtuse angles. By proceeding in a similar way, it may be shewn that BC cannot be otherwise than equal to EF. If ACB, DFE be both right angles: the case falls under Euc. 1. 26. 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. When a straight line intersects two other straight lines, two pairs of alternate angles are formed by the lines at their intersections, as in the figure, Bef, EFC are alternate angles as well as the angles AEF, EFD. Parallelism admits of being regarded as reciprocal: if AB be parallel to CD, CD is parallel to AB. 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 Euc. 1. 17; 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 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 Euc. 1. 27. 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 Euc. 1. 30; and is suggested by the diagram in the Commentary of Proclus on Euc. 1. 29. 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 straight 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 ypaμμal wapáλλnλoi, suggests the exact idea of such lines. 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 Colonel Thompson's "Geometry without Axioms." With respect to the different proposals made for the amendment of Euclid's method of treating the subject of parallel straight lines, it may be observed, that they all consist in setting out either with a different or a modified result from that of Euclid,—all true, and more or less obvious to the senses. Euclid has discussed the elementary properties of triangles, or of two lines which meet one another and are intersected by a third line, before he has entered upon the discussion of the properties of two lines which do not meet when they are intersected by a third line. The principal objection to Euclid's method of treating the subject of parallel lines, is the assumption of one truth as an axiom, which forms the converse of a theorem which he has demonstrated as the seventeenth proposition of the first book. Almost every writer on the subject admits the necessity of assuming some positive property of parallel lines as the basis of the reasonings on such lines: and that amendment of Euclid's method would seem to be the best, which simply supplies a defect, and leaves the so-called twelfth axiom to assume its rightful position as a theorem, and to fall into its proper place after the seventeenth proposition. Two straight lines in the same plane which do not meet, when produced, may be convergent or divergent with respect to each other, according to the directions in which both lines are produced; or, when produced in either direction, they may be neither divergent nor convergent. When a third line falls upon two straight lines and makes the two interior angles on one side of it less than two right angles; on that side of the line, the two straight lines are convergent, and will, if produced far enough, meet one another, as it is stated in the so-called twelfth axiom. On the other side of the line, the two interior angles are greater than two right angles, and the two straight lines are divergent, and will never meet, how far soever they may be produced. The limiting position of the two straight lines, is, when they are neither convergent nor divergent, that is, when they do not meet when produced in either direction; and such lines are then said to be parallel to one another. If the two parallel lines be intersected by a third line, the following properties exist respecting the angles, whether the intersecting line be perpendicular or be not perpendicular to either of the parallel lines. (1) The two interior angles on each side of the intersecting line are equal to two right angles. Euc. 1. 28. (2) The alternate angles on each side of the intersecting line are equal to one another. Euc. 1. 27. (3) The exterior angles are equal to their corresponding interior angles on the same side of the intersecting line. Euc. 1. 28. If the intersecting line be perpendicular to one of the parallel lines; it is also perpendicular to the other: and (4) The perpendicular distance between the two lines is always the same. If it has been correctly stated, that all axioms are in reality theorems assumed without proof, and that all demonstrated truths must depend on some truths assumed or admitted to be true, not necessarily truths first discovered, but truths the most simple and which arise directly from the subject of the definitions; the doctrine of parallel lines may be legitimately treated by assuming some one of the positive properties of such lines as the basis for demonstrating their other properties. Any one of the four positive properties just stated may be assumed as the foundation of the theory of parallel lines, and that theory may be made to depend on the distance between the parallel lines, or on some of the angles made by any intersecting line. If the former assumption be adopted: does it not involve that lines which are perpendicular to one of the parallel lines, are also perpendicular to the other, as well as that all such distances are equal? This would require more to be taken for granted, than would be necessary by assuming the equality of the exterior and interior angles, or either of the two remaining properties respecting the angles which the parallels make with any intersecting line. On the whole, perhaps, the axiom adopted by Playfair in proving Prop. XXIX, is the least objectionable of all the methods proposed for dispensing with the twelfth axiom, as an axiom, and letting that axiom take its place as the converse ́of Prop. XVII. Prop. xxx. In the diagram, the two lines AB and CD are placed one on each side of the line EF: the proposition may also be proved when both AB and CD are on the same side of Ef. Prop. XXXII. From this proposition, it is obvious that if one angle of a triangle be equal to the sum of the other two angles, that angle is a right angle, as is shewn in Euc. III. 31, and that each of the angles of an equilateral triangle, is equal to two-thirds of a right angle, as it is shewn in Euc. Iv. 15. Also, if one angle of an isosceles triangle be a right angle, then each of the equal angles is half a right angle, as in Euc. II. 9. The three angles of a triangle may be shewn to be equal to two right angles without producing a side of the triangle, by drawing through any angle of the triangle a line parallel to the opposite side, as Proclus has remarked in his Commentary on this proposition. It is manifest from this proposition, that the third angle of a triangle is not independent of the sum of the other two; but is known if the sum of any two is known. Cor. 1 may be also proved by drawing lines from any one of the angles of the figure to the other angles. If any of the sides of the figure bend inwards and form what are called re-entering angles, the enunciation of these two corollaries will require some modification. As Euclid has given no definition of re-entering angles, it may fairly be concluded, he did not intend to enter into the proofs of the properties of figures which contain such angles. Prop. XXXIII. The words "towards the same parts" are a necessary restriction: for if they were omitted, it would be doubtful whether the extremities A, C, and B, D were to be joined by the lines AC and BD; or the extremities A, D, and B, C, by the lines AD and BC. Prop. xxxiv. If the other diameter be drawn, it may be shewn that the diameters of a parallelogram bisect each other, as well as bisect the area of the parallelogram. And any straight line drawn through the bisection of the diagonals, also bisects the parallelogram. If the parallelogram be right-angled, the diagonals are equal; if the parallelogram be a square or a rhombus, the diagonals bisect each other at right angles. The converse of this Prop., namely, "If the opposite sides or opposite angles of a quadrilateral figure be equal, the opposite sides shall also be parallel; that is, the figure shall be a parallelogram," is not proved by Euclid. Prop. xxxv. The latter part of the demonstration is not expressed very intelligibly. Simson, who altered the demonstration, seems in fact to consider two trapeziums of the same form and magnitude, and from one of them, to take the triangle ABE; and from the other, the triangle DCF; and then the remainders are equal by the third axiom: that is, the parallelogram ABCD is equal to the parallelogram EBCF. Otherwise, the triangle, whose base is DE, (fig. 2.) is taken twice from the trapezium, which would appear to be impossible, if the sense in which Euclid applies the third axiom, is to be retained here. It may be observed, that the two parallelograms exhibited in fig. 2 partially lie on one another, and that the triangle whose base is BC is a common part of them, but that the triangle whose base is DE is entirely without both the parallelograms. After having proved the triangle ABE equal to the triangle DCF, if we take from these equals (fig. 2.) the triangle whose base is DE, and to each of the remainders add the triangle whose base is BC, then the parallelogram ABCD is equal to the parallelogram EBCF. In fig. 3, the equality of the parallelograms ABCD, EBCF, is shewn by adding the figure EBCD to each of the triangles ABE, DCF. In this proposition, the word equal assumes a new meaning, and is no longer restricted to mean coincidence in all the parts of two figures. Prop. xxxvIII. In this proposition, it is to be understood that the bases of the two triangles are in the same straight line. If in the diagram the point E coincide with C, and D with A, then the angle of one triangle is supplemental to the other. Hence the following property :-If two triangles have two sides of the one respectively equal to two sides of the other, and the contained angles supplemental, the two triangles are equal. 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. Prop. xxxix. If the vertices of all the equal triangles which can be described upon the same base, or upon the equal bases as in Prop. 40, be joined, the line thus formed will be a straight line, and is called the locus of the vertices of equal triangles upon the same base, or upon equal bases. A locus in plane Geometry is a straight line or a plane curve, every point of which and none else satisfies a certain condition. With the exception of the straight line and the circle, the two most simple loci; all other loci, perhaps. including also the Conic Sections, may be more readily and effectually investigated algebraically by means of their rectangular or polar equations. 1 Prop. XLL. The converse of this proposition is not proved by Euclid; viz.. If a parallelogram is double of a triangle, and they have the same base, or equal, bases, upon the same straight line, and towards the same parts, they shall be. between the same parallels. Also, it may easily be shewn that if two equal: triangles are between the same parallels; they are either upon the same base,, or upon equal bases. ་་ Prop. XLIV. A parallelogram described on a straight line is said to bę applied to that line. Prop. XLV. The problem is solved only for a rectilineal figure of four sides.. |