two angles of the one equal to two angles of the other, each to each; and one side equal to one side, viz. either the sides adjacent to the equal angles, or the sides opposite to the equal angles in each; then shall the other sides be equal, each to each; and also the third angle of the one to the third angle of the other. XXVII. If a straight line falling upon two other straight lines makes the alternate angles equal to one another, these two straight lines are parallel. XXVIII. If a straight line falling upon two other straight lines makes the exterior angle equal to the interior and opposite upon the same side of the line; or makes the interior angles upon the same side together equal to two right angles; the two straight lines are parallel to one another. XXIX. If a straight line fall upon two parallel straight lines, it makes the alternate angles equal to one another; and the exterior angle equal to the interior and opposite angle upon the same side; and likewise the two interior angles upon the same side together equal to two right angles. XXX. Straight lines which are parallel to the same straight line are parallel to one another. XXXII. If a side of any triangle be produced, the exterior angle is equal to the two interior and op. posite angles; and the three interior angles of every triangle are equal to two right angles. COR. 1. All the interior angles of any rectilineal figure are equal to twice as many right angles as the figure has sides, wanting four right angles. 2. All the exterior angles of any rectilineal figure are together equal to four right angles. XXXIII. The straight lines which join the extremities of two equal and parallel straight lines, towards the same parts, are also themselves equal and parallel. XXXIV. The opposite sides and angles of a parallelogram are equal to one another, and the diameter bisects it, that is, divides it into two equal parts. XXXV. Parallelograms upon the same base and between the same parallels, are equal to one another. XXXVI. Parallelograms upon equal bases, and between the same parallels, are equal to one another. XXXVII. Triangles upon the same base, and between the same parallels, are equal to one another. XXXVIII. Triangles upon equal bases, and between the same parallels, are equal to one another. XXXIX. Equal triangles upon the same base, and upon the same side of it, are between the same parallels. XL. Equal triangles on the same side of bases, which are equal and in

the same straight line, are between the same parallels. XLI. If a parallelogram and a triangle be upon the same base, and between the same parallels; the parallelogram is double of the triangle. XLIII. The complements of the parallelograms which are about the diameter of any parallelogram, are equal to one another. XLVII. In any right angled tri angle, the square which is described upon the side subtend. ing the right angle, is equal to the squares described upon the sides which contain the right angle. XLVIII. If the square described upon one of the sides of a triangle, be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle.

BOOK II.-DEFINITIONS. 1. Every right angled parallel. ogram, or rectangle, is said to be contained by any two of the straight lines which are about one of the right angles. 2. In every parallelogram, any of the parallelograms about a diameter, together with the two complements, is called a Gno.

mon.

PROP. I. If there be two straight lines, one of which is divided into any number of parts; the rectangle contained by the two straight lines is equal to the rectangles contained by the undivided line, and the several parts of the divided line. II. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts, are together equal to the square of the whole line. III. If a straight line be divided into any two parts, the rectangle contained by the whole and one of the parts, is equal to the rectangle contained by the two parts, together with the square of the foresaid part. IV. If a straight line be divided into any two parts, the square of the whole line is equal to the squares of the two parts, together with twice the rectangle contained by the parts. COR. From the demonstration, it is manifest that the parallelograms about the diameter of a square are likewise squares. V. If a straight line be divided into two equal parts, and also into two unequal parts; the rectangle contained by the unequal parts, together with the square of the line between the points of section, is equal to the square of half the line. COR. From this proposition it is manifest, that the difference of the squares of two unequal lines is equal to the rectangle contained by their sum and difference. VI. If a straight line be bisected, and produced to any point; the rectangle contained by the whole

line thus produced, and the part of it produced, together with the square of half the line bisected, is equal to the square of the straight line which is made up of the half and the part produced. VII. If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part. COR. Hence the sum of the squares of any two lines is equal to twice the rectangle contained by the lines together with the square of the difference of the lines. VIII. If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, together with the square of the other part, is equal to the square of the straight line which is made up of the whole and the first mentioned part. COR. 1. Hence, four times the rectangle contained by any two lines together with the square of their difference, is equal to the square of the sum of the lines. 2. From the demonstration it is also manifest, that the square of any line is quadruple of the square of half that line. IX. If a straight line be divided into two equal, and also into two unequal parts; the squares of the two unequal parts are together double of the square of half the line, and of the square of the line between the points of section. X. If a straight line be bisected, and produced to any point, the square of the whole line thus produced, and the square of the part of it produced, are together double of the square of half the line bisected, and of the square of the line made up of the half and the part produced. XII. In obtuse angled triangles, if a perpendicular be drawn from any of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted between the perpendicular and the obtuse angle. XIII. In every triangle the square of the side subtending any of the acute angles, is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perperdicular, let fall upon it from the opposite angle, and the acute angle. A. If one side of a triangle be bisected, the sum of the squares of the other two sides is double of the square of half the side

bisected, and of the square of the line drawn from the point of bisection to the opposite angle of the triangle. B. The sum of the squares of the diameters of any parallelogram is equal to the sum of the squares of the sides of the parallelogram. COR. From this demonstration, it is manifest that the diameters of every parallelogram bisect one another. BOOK III.-DEFINITIONS. A. The radius of a circle is the straight line drawn from the center to the circumference. 1. A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it. 2. Circles are said to touch one another, which meet, but do not cut one another. 3. Straight lines are said to be equally distant from the center of a circle, when the perpendiculars drawn to them from the center are equal. IV. And the straight line on which the greater perpendicular falls, is said to be farther from the center. B. An arch of a circle is any part of the circumference. 5. A segment of a circle is the figure contained by a straight line, and the arch which it cuts off. 6. An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the seg ment, to the extremities of the straight line which is the base of the segment. 7. And an angle is said to insist or stand upon the arch intercepted between the straight lines which contain the angle. 8. The sector of a circle is the figure contained by two straight lines drawn from the center, and the arch of the circumference between them. 9. Similar segments of a circle are those in which the angles are equal, or which contain equal angles.

PROP. II. If any two points be taken in the circumference of a circle, the straight line which joins them shall fall with. in the circle. III. If a straight line drawn through the center of a circle bisect a straight line in the circle, which does not pass through the center, it will cut that line at right angles; and if it cut it at right angles, it will bisect it. IV. If in a circle two straight lines cut one another, which do not pass through the center, they do not bisect each other. V. If two circles cut one another, they cannot have the same center. VI. If two circles touch one another internal. ly, they cannot have the same center. VII. If any point be taken in the diameter of a circle which is not the centre, of all the straight lines which can be drawn from it to the cir cumference, the greatest is that in which the centre is, and

the other part of that diameter is the least; and, of any others, that which is nearer to the line passing through the centre is always greater than one more remote from it and from the same point there can be drawn only two straight lines that are equal to one another, one upon each side of the shortest line. VIII. If any point be taken without a circle, and straight lines be drawn from it to the circumference, whereof one passes through the centre; of those which fall upon the concave circumference, the greatest is that which passes through the centre; and of the rest, that which is nearer to that through the centre is always greater than the more remote: but of those which fall upon the convex circumference, the least is that between the point without the circle, and the diameter; and of the rest, that which is nearer to the least is always less than the more remote and only two equal straight lines can be drawn from the point unto the circumference, one upon each side of the least. IX. If a point be taken within a circle, from which there fall more than two equal straight lines upon the cir cumference, that point is the centre of the circle. X. One circle cannot cut another in more than two points. XI. If two circles touch each other internally, the straight line which joins their centres being produced, will pass through the point of contact. XII. If two circles touch each other externally, the straight line which joins their centres will pass through the point of contact. XIII. One circle cannot touch another in more points than one, whether it touches it on the inside or outside. XIV. Equal straight lines in a circle are equally distant from the centre; and those which are equally distant from the centre are equal to one another. XV. The diameter is the greatest straight line in a circle; and of all others, that which is nearest the centre is always greater than the more remote; and the greater is nearer to the centre than the less. XVI. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn between that straight line and the circumference, from the extremity of the diameter, so as not to cut the circle. XVIII. If a straight line touch a circle, the straight line drawn from the center to the point of contact, is perpendicular to the line touching the circle. XIX. If a straight line touch a circle, and from the point of contact