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 4 may be to the line BC. It may be easily shewn from this proposition, that the difference of any two sides of a triangle is less than the third side. 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 lies 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 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, 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 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 ACÉ, 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. 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 EGR 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 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 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. 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 saine plane, and which neither recede from, nor approach to, each other." This includes the con ception 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 γραμμαὶ παράλληλοι, 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." 66 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. 11. 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. 11. 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 á 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 gives 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. 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. Prop. XLI. 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 be applied to that line. Prop. XLV. The problem is solved only for a rectilineal figure of four sides. If the given rectilineal figure have more than four sides, it may be divided into triangles by drawing straight lines from any angle of the figure to the opposite angles, and then a parallelogram equal to the third triangle can be applied to LM, and having an angle equal to E: and so on for all the triangles of which the rectilineal figure is composed. Prop. XLVI. The square being considered as an equilateral rectangle, its area or surface may be expressed numerically if the number of lineal units in a side of the square be given, as is shewn in the note on Prop. I., Book II. The student will not fail to remark the analogy which exists between the area of a square and the product of two equal numbers; and between the side of a square and the square foot of a number. There is, however, this distinction to be observed; it is always possible to find the product of two equal numbers, (or to find the square of a number, as it is usually called,) and to describe a square on a given line; but conversely, though the side of a given square is known from the figure itself, the exact number of units in the side of a square of given area, can only be found exactly, in such cases where the given number is a square number. For example, if the area of a square contain 9 square units, then the square root of 9 or 3, indicates the number of lineal units in the side of that square. Again, if the area of a square contain 12 square units, the side of the square is greater than 3, but less than 4 lineal units, and there is no number which will exactly express the side of that square: an approximation to the true length, however, may be obtained to any assigned degree of accuracy. Prop. XLVII. In a right-angled triangle, the side opposite to the right angle is called the hypotenuse, and the other two sides, the b se and perpendicular, according to their position. In the diagram the three squares are described on the outer sides of the triangle ABC. The Proposition may also be demonstrated (1) when the three squares are described upon the inner sides of the triangle: (2) when one square is described on the outer side and the other two squares on the inner sides of the triangle: (3) when one square is described on the inner side and the other two squares on the outer sides of the triangle. As one instance of the third case. If the square BE on the hypotenuse be described on the inner side of BC and the squares BG, HC on the outer sides of AB, AC; the point D falls on the side FG (Euclid's fig.) of the square BG, and KH produced meets CE in E. Let LA meet BC in M. Join DA; then the square GB and the oblong LB are each double of the triangle DAB, (Euc. 1. 41.); and similarly by joining_EA, the square HC and oblong LC are each double of the triangle EAC. Whence it follows that the squares on the sides AB, AC are together equal to the square on the hypotenuse BC. By this proposition may be found a square equal to the sum of any given squares, or equal to any multiple of a given square: or equal to the difference of two given squares. The truth of this proposition may be exhibited to the eye in some particular instances. As in the case of that right-angled triangle whose three sides are 3, 4, and 5 units respectively. If through the points of division of two contiguous sides of each of the squares upon the sides, lines be drawn parallel to the sides (see the notes on Book 11.), it will be obvious, that the squares will be divided into 9, 16 and 25 small squares, each of the same magnitude; and that the number of the small squares into which the squares on the perpendicular and base are divided is equal to the number into which the square on the hypotenuse is divided. Prop. XLVIII is the converse of Prop. XLVII. In this Prop. is assumed the Corollary that "the squares described upon two equal lines are equal," and the converse, which properly ought to have been appended to Prop. XLVI. The First Book of Euclid's Elements, it has been seen, is conversant with the construction and properties of rectilineal figures. It first lays down the definitions which limit the subjects of discussion in the First Book, next the three postulates, which restrict the instruments by which the constructions in Plane Geometry are effected; and thirdly, the twelve axioms, which express the principles by which a comparison is made between the ideas of the things defined. |