of the First Book of Euclid becomes a corollary to the last article; and may be cited in the demonstration of Prop. 27. Book I.; also the 29th Proposition of this Book may be proved by a reductio ad absurdum, without the help of the 12th axiom, which itself becomes a corollary to that proposition. Nothing seems to be wanting, to render the definition here substituted unobjectionable, but a proof, that if it be a property of two straight lines at any one point, it will also obtain at every other point of them. This proof has been given by Robert Simson in his Note upon E. 29. 1.; and it is made to depend only upon an axiom and upon the 4th and 8th Proposition of the First Book of Euclid. It has also been given, under a somewhat different form, by Borelli, in a single proposition, clearly and elegantly demonstrated; but the axiom borrowed from the Arabian mathematicians, which he premises, is, perhaps, less judiciously chosen, than that which is the foundation of Simson's proof. The great object of this substitution is to avoid the necessity of having recourse to the proposition, which Euclid has made his 12th axiom; but neither is his 35th definition itself the best that might be given; for, with the exception of the single supposition, that the two straight lines are in the same plane, it is wholly negative, and affords no practical test of the parallelism of two straight lines.

A modern editor of the Elements, of great learning and ability, has proposed the following axióm,

instead of the 12th :-"Two straight lines cannot be drawn through the same point parallel to the same straight line, without coinciding with one another." But this pre-supposes the 35th Definition of Euclid, and is, therefore, objectionable, inasmuch as that definition is itself objectionable. Besides, if it be thus stated, according to its true meaning,-"Two straight lines cannot be drawn through the same point, neither of which, when they are produced ever so far both ways, meets another straight line, given in position, but indefinite in length," it does not appear to have a much better claim to the title of axiom, than the assumption which it is intended to replace. Garnier has accordingly, in his Geometry, formally demonstrated this very proposition: It is, indeed, most certain, that the mind cannot conceive two straight lines in a state of infinite extension; which, however, it is led to attempt, by the 35th definition, and the axiom last considered.

PROP. XX.

(IX.)

The three sides of a triangle taken together, exceed the double of any one side, and are less than the double of any two sides.

(x.)

Any side of a triangle is greater than the difference between the other two sides.

(XI.)

Any one side of a rectilineal figure is less than the aggregate of the remaining sides..

(XII.)

The two sides of a triangle are together, greater than the double of the straight line which joins the vertex and the bisection of the base.

PROP. XXI.

(XIII.)

If a trapezium and a triangle stand upon the same base, and on the same side of it, and the one figure fall within the other, that which has the greater surface shall have the greater perimeter.

PROP. XXVI.

(XIV.)

If two right-angled triangles have the three angles of the one equal to the three angles of the

other, each to each, and if a side of the one be equal to the perpendicular let fall from the right angle upon the hypotenuse of the other, then shall a side of this latter triangle be equal to the hypotenuse of the former.

(xv.)

If the sides of any given equilateral and equiangular figure of more than four sides, be produced so as to meet, the straight lines, joining their several intersections, shall contain an equilateral and equiangular figure, of the same number of sides as the given figure.

PROP. XXVII.

(XVI.)

If two opposite sides of a quadrilateral figure be equal to one another, and the two remaining sides be also equal to one another, the figure is a parallelogram.

COR. 1. Hence may be deduced a practical method of drawing a straight line, through a given point, parallel to a given straight line.

COR. 2. A rhombus is a parallelogram.

PROP. XXIX.

(XVII.)

Every parallelogram which has one angle a right angle, has all its angles right angles.

(XVIII.)

To trisect a right angle; i. e. to divide it into three equal parts.

(XIX.)

Hence, to trisect a given rectilineal angle, which is the half, or the quarter, or the eighth part, and so on, of a right angle.

PROP. XXXI.

(xx.)

To find a point, in either of the equal sides of a given isosceles triangle, from which, if a straight line be drawn, perpendicular to that side, so as to meet the other side produced, it shall be equal to the base of the triangle.

(XXI.)

In the hypotenuse of a right-angled triangle, to find a point, the perpendicular distance of which