69. Def. 15. We have here a complete and satisfactory instance of the method of defining a species by means of the genus and special difference. (Art. 23, 24.) "A circle is a plane figure," it belongs to that class of figures, which have all their parts in the same plane, and consequently agrees in this general character with a triangle, a square, a polygon, an ellipsis, &c. it is" contained by one line called the circumference;" here we have a limitation whereby all such figures as are contained by more than one line, as the triangle, square, polygon, &c. are excluded; "and is such that all straight lines drawn from a certain point within the figure" (called in the next following definition" the centre") to the circumference, are equal to one another: this latter clause operates as an additional limitation, which excludes the ellipsis and all irregular curvilineal figures from the definition, because there is no point in either of those figures, from whence all the straight lines drawn to the circumference are equal. Here then we are informed, first, to what general class of figures a circle belongs, and secondly, by what it differs from every other figure of that class; whence the definition furnishes us with an adequate and precise idea of the figure called a circle.

70. Another definition, in substance the same as Euclid's, is this: "A circle is a figure generated (or formed) by a straight line revolving (or turning) in a plane about one of its extreme points, which remains fixed," the fixed point being the centre, and the line described by the revolving point the circumference.

71. The circumference of a circle is likewise called the periphery: it is sometimes improperly named the circle; a circle, in the proper acceptation of the term, means the space included within the circumference, and not the circumference exclusively.

72. To describe a circle with the compasses, you have only to fix one foot at the point where the centre is intended to be, and (the compasses being opened to a proper extent) turn the other foot quite round, and it will trace out the circumference. 73. After Def. 17. add the following, which is in continual use, viz. “ a radius, or semidiameter of a circle, is a straight line drawn from the centre to the circumference."

74. Def. 18, 19. Any part of a circle cut off by a straight line, is called a segment of a circle; if the straight line pass through the centre, it is a diameter, (Def. 17.) and divides the

circle into two equal segments, called semi-circles: but if the straight line which cuts the circle does not pass through the centre, it will divide the circle into two unequal segments, the greater of which is said to be "a segment greater than a semicircle," and the less "a segment less than a semi-circle." By the terms "segment of a circle," and "semi-circle," we are always to understand the space included between a part of the circumference and the straight line by which that part is cut off, unless the contrary be expressed.

75. Any part of the circumference is called an arc, and the straight line which joins the extremities of an arc, (or which divides the circle into two segments,) is called a chord, viz. it is the common chord of both the arcs into which it divides the whole circumference.

76. Def. 23. We have nothing to do professedly with polygons in the first book, yet since the definition is introduced, it may not be improper to observe, that a polygon, having all its sides and angles respectively equal, is called an equilateral, equiangular, or regular polygon. These figures are named according to their number of sides; thus,

77. Def. 24, 25, 26, 27, 28, and 29. Triangles are distinguished into three varieties with respect to their sides, and three with respect to their angles: the three varieties denominated from their sides, (as laid down in Def. 24, 25, and 26.) are equilateral, isosceles, and scalene; the latter, although defined here, does not occur under that name in any other part of the Elements. The three varieties which respect their angles, are rightangled, obtuse-angled, and acute-angled. Def. 27, 28, and 29.

78. Def. 30. A square, which according to this definition "has all its sides equal, and all its angles right angles," must evidently be just as wide as it is long; hence there can be no such thing as a long square, although we read of such a figure in some books ".

Euclid's definition of a square may be considered as faulty, for with the essential properties of a square he has incorporated an inference, which is the

79. Def. 31. Since the word oblong does not once occur in any subsequent part of the Elements, it should not have found a place here. The figure defined is a species of that which is called in the second book, and elsewhere, a rectangle.

80. Def. 35. In the definition of parallel lines as here laid down, Dr. Simson has improved on Euclid, and his definition is better adapted to the learner's comprehension than either of those approved by Wolfius, Boscovich, Thomas Simpson, D'Alembert, or Newton; the truth is, that no inference can be drawn from any definition hitherto given, sufficient to fix the doctrine of parallel lines on the firm basis of unobjectionable evidence °.

subject of the cor. to prop. 46. b. 1. It would be more strictly scientific to define a square to be "a four-sided figure having all its sides equal, and one of its angles a right angle;" for that "an equilateral four-sided figure is a parallelogram," and that "every parallelogram having one right angle has all its angles right angles," are plainly inferences from the definition given in this note, and that of a parallelogram, prop. 34. b. 1. the like observations extend to Def. 32. In both instances Euclid has abandoned his own plan, and transgressed a rule which ought never to be violated without absolute necessity; the departure is however justifiable in the present instance, as Euclid's definition will be more easily understood by a beginner than that which we have proposed.

• Having explained the definitions as they stand in Euclid, we may be allowed to remark, that a more methodical arrangement of them would be a desirable improvement; should any future Editor think this hint worth his attention and adopt it, it will be conducive to elegance, correctness, clearness, and simplicity, which are undoubtedly points of importance, especially at the beginning of the Elements. The alterations I would propose are as follow:

Def. 18. A segment of a circle is the figure contained by a straight line, and the circumference it cuts off.

19. If the straight line be a diameter, the segment is called a semi-circle. From the 20th to the 29th inclusive, may stand as at present.

30. Parallel straight lines are such as are in the same plane, and which, being produced ever so far both ways, do not meet.

31. A parallelogram is a four-sided figure, of which the opposite sides are parallel.

32. The diameter or diagonal of a parallelogram is a straight line which joins any two of its opposite angles.

33. A rhombus is a parallelogram which has all its sides equal, but its angles are not right angles.

On the Postulates.

81. A postulate, as we have before observed, is a self-evident practical proposition: on this subject Mr. Ludlam very justly remarks, that "Euclid does not here require a practical dexterity in the management of a ruler and pencil, but that the postulates are here set down that his readers may admit the possibility of what he may hereafter require to be done.” On this we remark, that our conviction of the possibility of any operation depends on our having actually performed it in some particular instance ourselves, or known that it has been performed by others; having thus satisfied itself of the possibility in particular instances, the mind immediately perceives that the possibility extends to every instance, or that the operation is true in general. On these considerations it has been affirmed, that "the mathematical sciences are sciences of experiment and observation, founded solely upon the induction of particular facts, as much so as mechanics, astronomy, optics, or chemistry P." This doctrine, to its fullest extent, it would perhaps be unsafe to adopt.

82. In applying the postulates, we proceed in an order the converse of that laid down in the preceding article: we admit what is affirmed in the postulate to be true in general, i. e. in all cases; and since it is true in all cases, it follows as a necessary inference, that it is true in the particular case under consideration. We will now begin to exemplify the use of the mathe

34. A rhomboid is a parallelogram of which all its sides are not equal, nor any of its angles right angles.

35. A rectangle is a parallelogram which has all, its angles right angles (or which has one of its angles a right angle; see the foregoing note.)

36. A square is a rectangle which has all its sides equal.

37. All other four-sided figures besides these are called trapeziums. Note. A trapezium which has two of its sides parallel, is sometimes called a trapezoid, and a straight line joining the opposite angles of a trapezium is called its diagonal.

The definitions preceding the 18th might stand as they do at present, if instead of the first definition, that which we have proposed (see Art. 52.) were adopted.

p The postulates prefixed to the Elements are in number (as they ought to be) the fewest possible; for, as Sir Isaac Newton observes, "postulates are principles which Geometry borrows from the arts, and its excellence consists in the paucity of them." The postulates of Euclid are all problems derived from the mechanics. Ingram.

matical instruments, to afford the student an opportunity of practical as well as mental improvement.

83. Postulate 1. If it be granted, that "a straight line may be drawn from any one point to any other point," it follows as an evident consequence, that a straight line can be drawn from the point A to the point B. Lay a straight scale or ruler, so that its edge may touch the two proposed points A and B, then with a pen or pencil draw along the edge of the scale or ruler a line from A to B, and what was granted in general will in this particular instance be performed.

84. Post. 2. To produce a line means to lengthen it. A straight line of two inches in length, may according to this postulate be produced until it is three, four, five, or more inches in length. Lay the edge of your scale touching every point of the given line, and with the pencil or pen, as before, draw the line to the length proposed.

85. Post. 3. Extend the points of the compasses to the required distance, then with one foot fixed on the given point as a centre, let the other be turned completely round on the paper, and it will describe the circle required.

On the Axioms.

86. An axiom is a self-evident theoretical proposition, which neither admits of, nor requires proof. Axioms evidently depend in the first instance on particular observation, from whence the mind intuitively perceives their truth in general: like the postulates, these general truths being previously laid down and acknowledged, are applied to the proof of the demonstrable propositions which follow.

87. Axioms 1, 2, 3, 4, 5, 6, 7, 9, and 10, are too plain to require illustration; the 10th is what is usually called an identical proposition, amounting to no more than this, namely, that "all right angles are right angles."

88. Ax. 8. Should the learner feel disposed to hesitate at this axiom, he may be informed, that every one readily admits its truth in practical matters; a farmer who has two quantities of corn, each of which exactly fills his bushel, would be surprised if any one should deny that these two quantities are equal to each other.

89. The 12th axiom, as it is called, is not properly an axiom, but a proposition which requires proof; the learner, if he can