XXVI. A scalene triangle is that which has three unequal sides. XXVII. A right-angled triangle is that which has a right angle. XXVIII. An obtuse-angled triangle is that which has an obtuse angle. XXIX. An acute-angled triangle is that which has three acute angles. XXX. Of four-sided figures, a square is that which has all its sides equal, and all its angles right angles. XXXI. An oblong is that which has all its angles right angles, but has not all its sides equal. XXXII. A rhombus is that which has all its sides equal, but its angles are not right angles. XXXIII. A rhomboid is that which has its opposite sides equal to one another, but all its sides are not equal, nor its angles right angles. コロ XXXIV. All other four-sided figures besides these are called Tra peziums. XXXV. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet. (A.) A trapezium having two, only, of its sides parallel to one another is now called a trapezoid. POSTULATES. LET it be granted that a straight line may be drawn from any one point to any other point. II. That a terminated straight line may be produced to any length in a straight line. III. That a circle may be described from any centre, at any distance from that centre. IV. [Ax. XI.] ས. And that if a straight line meets two straight lines, so as to make the two interior angles on the same side of it, taken together, less than two right angles, these straight lines, being continually produced, shall at length meet upon that side on which are the angles less than two right angles. AXIOMS. THINGS which are equal to the same are equal to one another. II. If equals be added to equals, the wholes are equal. III. If equals be taken from equals, the remainders are equal. IV. If equals be added to unequals, the wholes are unequal. V. If equals be taken from unequals, the remainders are unequal. VI. Things which are doubles of the same are equal to one another. VII. Things which are halves of the same are equal to one another. VIII. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another. N.B. The term 'equivalent' is used in this work to denote equality of magnitude, in surface or volume, when figures (plane or solid) on being applied to one another will not coincide, or occupy the same space: the term 'equal" is used when, on being applied, they will coincide. IX. The whole is greater than its part. X. Two straight lines cannot inclose a space. XI. All right angles are equal to one another. Remarks on the Definitions, &c. of the First Book. THE properties of geometrical magnitudes depend upon the definitions of the objects contemplated and upon certain abstractions of the mind, as position and straightness, which are among the qualities attributed to those objects: the investigations of the properties depend also on certain propositions which are assumed to be self-evident; and the impossibility of expressing abstract qualities otherwise than by negations is the cause that there is felt to be a want of precision in the language employed at the very commencement of the elements of geometry. Euclid defines a point by saying that it has no parts or no magnitude (Def. I.): circumstances which may with equal correctness be predicated of non-entity; Professor Playfair designates it 'that which has position without magnitude,' which is nearly the definition said to have been given by Pythagoras (a monad in position), and which is still defective, since we have no idea of position, except as a part of space, which is, or may be, occupied by magnitude. Euclid's definition of a straight line, that which lies evenly, or equally, between its extreme points,' must be admitted to be very obscure. To lie evenly between the extreme points may signify that the line does not tend towards the parts of space on either of its sides; which is the sense of Plato's definition: and to lie equally between the extreme points may signify that the line divides the infinite space on the opposite sides of it into two equal parts. Archimedes defines a straight line to be the shortest distance between its extremities: and it is evident that none of these definitions has the precision which should characterise an elementary principle. The like observation may be made on Euclid's seventh definition, "A plane superficies is that which lies evenly between its boundary lines. That which Dr. Simson has substituted for it (in the text) is certainly more distinct; it has not, however, the propriety of corresponding to the previous definition of a straight line. But the circumstance which has given most trouble to those geometers who have attempted to improve the Elements of Euclid arises out of the properties of parallel lines. In the twenty-ninth proposition of the first book, the proof that 'if a straight line fall upon two parallel straight lines it will make the alternate angles equal to one another,' &c. is made to depend upon an assumption that, if a line falling upon two other lines makes with them the interior angles on the same side less than two right angles, those lines must meet if produced. Now this is evidently a theorem which ought to have been previously demonstrated, whereas Euclid has made it one of his postulates (Post. V.), and he thus leaves an elementary truth without direct support; for there exists no evidence of its certainty, except that which may be drawn from a converse of the twenty-eighth proposition of the first book, where it is shown that if a straight line falls upon two other straight lines, and makes the sum of the two interior angles upon the same side equal to two right angles, the two straight lines shall be parallel. Numerous efforts have, at various times, been made to obtain a direct proof of this postulate, and also to change the assumption in the twentyninth proposition for one more simple; but it may be said that every such attempt which has been founded on the elementary geometry only contains, in some manner, the defect which it was intended to obviate. Professor Playfair and other mathematicians have recommended that, for the postulate, there should be substituted the axiom, Two lines which intersect one another cannot be parallel to a third;' which has the advantage over Euclid's postulate in simplicity only, while Dr. Simson has made the postulate itself an axiom, admitting at the same time that it is not a self-evident truth; and, in fact, in his note on the twenty-ninth proposition, he has given a demonstration of it in five theorems. This change of a postulate into an axiom, which cannot be said to carry in itself the evidence of its truth, and which therefore does not possess a character essential to all axioms, must be considered as unsatisfactory; and it will probably be admitted that, since every effort to remove the difficulty has hitherto been without success, the most legitimate course to be pursued is to leave the proposition in the form of a postulate. The place in which it ought properly to be introduced must, however, be understood to be immediately after the twenty-eighth proposition of the first book. Euclid's fourth postulate has very properly been placed by Dr. Simson at the end of the axioms (Ax. XI.) Dr. Simson considers that the eighth definition was intended to include an angle formed in a plane by the meeting of two curve lines, or of a straight line and a curve, as well as that formed by two straight lines, which is the subject of the ninth definition; and the two first kinds of plane angles being useless, Dr. Simson has distinguished the eighth by double inverted commas. For the reason last mentioned, the thirteenth definition is so marked: the nineteenth, properly belongs to the third book, where it is repeated. PROPOSITION I. PROBLEM. To describe an equilateral triangle upon a given finite straight line. Let AB be the given straight line; it is required to describe an equilateral triangle upon it. From the centre A, at the dis- (D tance AB, describe (3. Postulate) the circle BCD, and from the centre B, at the distance BA, describe the A Ө B L circle ACE; and from the point C, in which the circles cut one another, draw the straight lines (1. Post.) CA, CB to the points A, B: ABC shall be an equilateral triangle. Because the point A is the centre of the circle BCD, AC is equal (15. Definition) to AB; and because the point B is the centre of the circle ACE, BC is equal to BA: but it has been proved that CA is equal to AB; therefore CA, CB are each of them equal to AB: but things which are equal to the same are equal to one another (1. Axiom); therefore CA is equal to CB; wherefore CA, AB, BC are equal to one another; and the triangle ABC is therefore equilateral, and it is described upon the given straight line A B. Which was required to be done. In practice; instead of describing complete circles about A and B as centres, arcs or portions of the circumferences, of any convenient length, are described with a radius or semidiameter equal to AB, so as to intersect each other in c; and then the lines CA, CB are drawn, as in the text. PROP. II. PROB. FROM a given point to draw a straight line equal to a given straight line. Let A be the given point, and BC the given straight line; it is required to draw from the point a a straight line equal |