CONTENTS.

DEFINITION I. A rectilineal (or right-lined) Angle is that which is formed by two right lines meeting at the same point, but not lying in the same right line, 2.

DEF. II. Figures are bounded portions of space, 3.

DEF. III. A plane figure is a plane surface bounded by one or more lines, ibid.

DEF. IV. A plane rectilineal figure is a plane surface bounded by right lines, ibid.

DEF. V. A rectilineal triangle is a plane figure bounded by three right lines, ibid.

DEF. VI. If one right line, standing upon another, make the adjacent angles equal to one another, each of these angles is called a right angle, and the right line which stands upon the other is called a perpendicular to it, 12.

DEF. VII. An angle greater than a right angle is called an obtuse angle; and an angle less than a right angle is called an acute angle, 13. DEF. VIII. If two right lines cut each other so as to form four angles, each opposite pair are called Vertically opposite angles, 16.

DEF. IX. Two right lines are said to be equally distant from one another when any two points whatsoever in the one not the greater, and any two equally remote points in the other, being taken, the right lines

PROB. I. To describe an equilateral triangle on a given finite right line, 5.

PROB. II. From a given point to draw a right line equal to a given finite right line, 6.

PROB. III. From the greater of two given right lines to cut off a part equal to the less, 7.

PROB. IV. To divide a given rectilineal angle into two equal parts, 10.

PROB. V. To divide a given finite right line into two equal parts, 11. PROB. VI. To draw a perpendicular to a given right line, from a given point without it, 11.

PROB. VII. To draw a perpendicular to a given right line, from a point in it, 12.

PROB. VIII. At a given point in a given indefinite right line, to draw a right line, making, with the given one, an angle equal to a given angle,

13.

PROB. IX. Through a given point outside a given right line to draw a right line parallel to the given one, 19.

which join each opposite pair of points towards the same hand are equal to each other, 16.

DEF. X. Parallel right lines are those which are equally distant from each other, 17.

DEF. XI. A parallelogram is a four-sided rectilineal figure, each pair of whose opposite sides are parallel, 22.

DEF. XII. Taking any side of a triangle as base, the perpendicular from the vertex to the base (produced if necessary) is called the Altitude of the triangle. Also, taking any side of a parallelogram as base, the perpendicular from any point in the opposite side, on the base (produced if necessary), is called the Altitude of the parallelogram, 25.

DEF. XIII. A square is a parallelogram whose two adjacent sides are equal, and any of whose angles is a right angle, 28.

DEF. XIV. A Circle is a plane figure bounded by one line, such, that all right lines drawn from it to one and the same point are equal to each other, 41.

DEF. XV. In a circle the bounding line is called the Circumference, and the point to which the equal lines are drawn from the circumference is called the Centre, ibid.

DEF. XVI. Any right line terminated both ways in a circle is called a Chord of the circle, or of the arch it cuts off, ibid.

DEF. XVII. A right line which meets a circle in two points, but is not terminated in both, is called a Secant, 44.

DEF. XVIII. A right line drawn from the centre of a circle and terminated in the circumference, is called a Radius, ibid.

DEF. XIX. A right line drawn through the centre of a circle and terminated both ways in the circumference, is called a Diameter, ibid.

DEF. XX. A right line which, however produced, meets a circle in but one point, is called a Tangent to that circle, 45.

DEF. XXI. Right lines are said to be equally distant from the centre

PROB. X. On a given right line to describe a square, 28.
PROB. XI. To find the centre of a given circle, 42.

PROB. XII. From a given point without a given circle, to draw a right line which shall be a tangent to the circle, 45.

PROB. XIII. A segment of a circle being given, to describe the circle of which it is the segment, 54.

PROB. XIV. To divide a given arch of a circle into two equal parts, 63. PROB. XV. To divide a given finite right line into any number of equal parts, 75.

PROB. XVI. To divide a right line into two parts which shall have the same ratio as the two parts of a given divided line, 84.

PROB. XVII. To cut off from a given right line a part which shall have a certain given ratio to the whole line, ibid.

PROB. XVIII. To find a fourth proportional to three given finite lines, 85.

PROB. XIX. To find a third proportional to two given finite right lines, ibid.

PROB. XX. To find a mean proportional between any two given finite right lines, 86.

of a circle when the perpendiculars drawn to them from the centre are equal. And one right line is said to be farther from the centre than another when the perpendicular on the former is greater than that on the latter, 47.

DEF. XXII. If from a point outside a circle two tangents be drawn to the circle, and the points of contact be joined by a right line, that part of the circumference lying within the triangle thus formed is called the Convex, and that part of the circumference lying without this triangle is called the Concave part of the circumference, with respect to the given point, 51.

DEF. XXIII. A rectilineal figure is said to be inscribed in a circle when the vertices of all its angles are situated in the circumference of that circle, 53.

DEF. XXIV. If any geometrical figure or magnitude be divided into two or more parts, these parts are called Segments, 54.

DEF. XXV. When an angle has its vertex in an arch of a circle, and its sides terminated in the extremities of that arch, this angle is called the Angle in a segment of the circle, 55.

DEF. XXVI. The segments into which a circle is divided by its diameter are called Semicircles, 56.

DEF. XXVII. When one circle meets another in two points, it is said to cut it, 58.

DEF. XXVIII. Circles which meet in but one point are said to touch each other, 59.

DEF. XXIX. The point where two circles touch is called the point of contact, ibid.

DEF. XXX. Equal circles are those which, if applied centre to centre, would exactly coincide with each other, 60.

DEF. XXXI. A Rectangle is a right-angled parallelogram, 64.

DEF. XXXII. When one magnitude exactly equals another magnitude added to itself an integral number of times, the greater magnitude is called a Multiple of the lesser; and the lesser a Submultiple of the greater, 71.

DEF. XXXIII. When two multiples contain their respective submultiples exactly an equal number of times, they are called Equi-multiples. And when two submultiples are contained in their respective multiples exactly an equal number of times they are called Equi-submultiples, ibid.

PROB. XXI. Given three finite lines, of which any two are greater than the third, to construct a triangle of which the sides shall be respectively equal to the given lines, 96.

98.

PROB. XXII. To find a square equal to the sum of two squares, 97. PROB. XXIII. To find a square equal to the difference of two squares,

PROB. XXIV. To draw a tangent at any given point in the circumference of a circle, 106.

PROB. XXV. To divide a given right line into any number of parts which shall have the same ratio to each other as the parts of another given divided right line, 113.

PROB. XXVI. To make a square equal to a given rectangle, 114.

b

DEF. XXXIV. Ratio is the mutual relation of two quantities of the me kind to each other, with respect to magnitude, 73.

same

DEF. XXXV. Two quantities are said to have the same ratio to each other as two other quantities have, when every submultiple of the first quantity is contained in the second the same integral number of times that an equi-submultiple of the third quantity is contained in the fourth, 74.

DEF. XXXVI. When a certain part of one magnitude has the same ratio to a certain part of another magnitude, as a second part of this other has to a second part of the first, the proportion subsisting between these four parts is called, for brevity's sake, reciprocal proportion, 77.

DEF. XXXVII. When two parallelograms have an angle in the one equal to an angle in the other, they are called equi-angular parallelograms, ibid.

DEF. XXXVIII. If there be a series of magnitudes of the same kind, in which the 1st has to the 2d the same ratio as the 2d has to the 3d; and the 2d has to the 3d the same ratio as the 3d has to the 4th; and the 3d has to the 4th the same ratio as the 4th has to the 5th; and so onthen, these magnitudes are said to be in continued proportion, 80.

DEF. XXXIX. When three magnitudes are proportionals, the first is said to have to the third a ratio duplicate of that which it has to the second, 91.

DEF. XL. If through a given point in the diagonal of a parallelogram two right lines be drawn parallel respectively to two adjacent sides of the parallelogram, so as to make four parallelograms,-then, those two through which the diagonal runs are called parallelograms about the diagonal, and the remaining two are called complements of the parallelograms about the diagonal, 100.

AXIOM 1. Two right lines cannot completely enclose a space, 4.

Ax. 2. Magnitudes which coincide exactly with each other are equal, 5. Ax. 3. Things which are equal to the same thing are equal to each other, 6.

Ax. 4. If from equal things we take away equal things, respectively, the remainders will be equal, ibid.

Ax. 5. The whole is greater than its part, 9.

Ax. 6. If to equal things we add equal things, respectively, the wholes will be equal, 11.

Ax. 7. The halves of equal things are equal, 24.

Ax. 8. The doubles of equal things are equal, 26.

Ax. 9. If from unequal things we take equal things, the remainders are unequal, 49.

ARTICLE 1. If there be two triangles which have two sides of the one equal, respectively, to two sides of the other; and likewise the angles contained by those sides equal to one another :-then, the bases or third sides of these triangles are also equal, 3.

ART. 2, In such triangles as above described, those angles at the bases which are opposite to equal sides, are respectively equal, 5.

ART. 3. Such triangles as above described are equal, in every respect, to each other, ibid.

ART. 4. A triangle which has two of its sides equal, has also its angles, opposite the equal sides, equal, 8.

ART. 5. A triangle which has two of its angles equal, has also its sides opposite the equal angles equal, 9.

ART. 6. If there be two triangles which have all the sides of the one equal, respectively, to all the sides of the other, then the angles, which in these triangles are opposite to equal, sides, are also equal, ibid.

ART. 7. Such triangles as are described in the preceding Article are equal, in every respect, to each other, 11.

ART. 8. All right angles are equal, 13.

ART. 9. When a right line meeting another right line makes angles with it, these angles are together equal to two right angles, 15.

ART. 10. When two right lines meet another at the same point, but at different sides, and make angles with it which are together equal to two right angles, those right lines are in one continued right line, ibid.

ART. 11. If two right lines intersect one another, the vertically opposite angles are equal, 16.

ART. 12. If a right line intersect two parallel right lines, it makes the alternate angles equal to each other, 17.

ART. 13. If a right line intersect two parallel right lines, it makes the two internal angles on the same side of the intersecting line together equal to two right angles, 18.

ART. 14. If a right line intersect two parallel right lines, it makes cach external angle equal to the farther internal angle on the same side of the intersecting line, ibid.

ART. 15. If a right line intersect two right lines, and make the alternate angles equal to each other, these two latter right lines are parallel, 19.

ART. 16. If a right line intersect two right lines, and make the external angle equal to the farther internal angle at the same side of the intersecting line, these two latter right lines are parallel, 20.

ART. 17. If a right line intersect two right lines, and make the two internal angles at the same side of the intersecting line together equal to two right angles, these two latter right lines are parallel, ibid.

ART. 18. If a right line intersect two parallel right lines, and another right line be drawn parallel to the intersector from any point in either of the parallels, it will meet the other, if produced sufficiently; and its length between the parallels will be equal to the length of the intersector between the parallels, ibid.

ART. 19. Right lines which join the adjacent extremities of two equal and parallel lines are themselves equal and parallel, 22.

ART. 20. The opposite sides of a parallelogram are equal, ibid.
ART. 21. The opposite angles of a parallelogram are equal, 23.

ART. 22. A parallelogram is divided into two equal parts by its diagonal, ibid.

ART. 23. Parallelograms on the same base, and between the same parallels, are equal, ibid.

ART. 24. Parallelograms on equal bases and between the same parallels are equal, 24.

ART. 25. Triangles upon the same base and between the same parallels are equal, ibid. b 2