Elementary or Common Geometry, is that which is employed in the consideration of right lines and plane surfaces, with the solids generated from them. And the

Higher or Sublime Geometry, is that which is employed in the consideration of curve lines, conic sections, and the bodies formed of them. This part has been chiefly cultivated by the moderns, by help of the improved state of algebra, and the modern analysis or fluxions.

The standard author on the elements of geome. try is Euclid. See EUCLID and ELEMENTS. The definitions and propositions of his first six, and eleventh and twelfth books, are as below.

Book I. Def. 1.-A point is that which hath no magnitude.

2. A line is length without breadth.

3. The extremities of a line are points.

4. A straight line is that which lies evenly between its extreme points.

5. A superficies is that which hath only length and breadth.

6. The extremities of a superficies are lines. 7. A plane superficies is that in which any two points being taken, the straight line between them lies wholly in that superficies.

8. "A plane angle is the inclination of two lines to one another in a plane, which meet together, but are not in the same direction."

9. A plane rectilineal angle is the inclination of two straight lines to one another, which meet together, but are not in the same straight line.

10. When a straight line standing on another straight line makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called a perpendicular to it.

11. An obtuse angle is that which is greater than a right angle.

12. An acute angle is that which is less than a right angle.

13. "A term or boundary is the extremity of any thing."

14. A figure is that which is inclosed by one or more boundaries.

15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference, are equal to one another.

16. And this point is called the centre of the circle.

17. A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference.

18. A semicircle is the figure contained by a diameter and the part of the circumference cut off by the diameter.

19. "A segment of a circle is the figure con"tained by a straight line, and the circumference it cuts off."

20. Rectilineal figures are those which are contained by straight lines.

21. Trilateral figures, or triangles, by three straight lines.

22. Quadrilateral, by four straight lines. 23. Multilateral figures, or polygons, by more than four straight lines.

24. Of three sided figures, an equilateral triangle is that which has three equal sides.

25. An isosceles triangle, is that which has only two sides equal.

26. A scalene triangle, is that which has three unequal sides.

27. A right angled triangle, is that which has a right angle.

28. An obtuse angled triangle, is that which has an obtuse angle.

29. An acute angled triangle, is that which has three acute angles.

30. Of four sided figures, a square is that which has all its sides equal, and all its angles right angles.

31. An oblong, is that which has all its angles right angles, but has not all its sides equal.

32. A rhombus, is that which has all its sides equal, but its angles are not right angles.

33. 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.

34. All other four sided figures besides these, are called trapeziums.

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

Postulates.-1. Let it be granted that a straight line may be drawn from any one point to any other point.

2. That a terminated straight line may be produced to any length in a straight line.

3. And that a circle may be described from any centre, at any distance from that centre.

Axioms.-1. Things which are equal to the same are equal to one another.

2. If equals be added to equals, the wholes are equal.

5. if equais be taken from equals, the remainders are equal.

4. If equals be added to unequals, the wholes are unequal.

5. If equals be taken from unequals, the remainders are unequal.

6. Things which are double of the same, are equal to one another.

7. Things which are halves of the same, are equal to one another,

8. Magnitudes which coincide with one another, that is, which exactly fill the same space, are equal to one another.

9. The whole is greater than its part.

10. Two straight lines cannot inclose a space. 11. All right angles are equal to one another. 12. "If a straight line mects 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 which are less than two right angles. See the notes on Prop. XXIX. of Book L.”

Prop. I. Prob. To describe an equilateral triangle upon a given finite straight line.

Prop. II. Prob. From a given point to draw a straight line equal to a given straight line. Prop. III. Prob. From the greater of two given straight lines to cut off a part equal to the less.

Prop. IV. Theor. If two triangles have two sides of the one equal to two sides of the other, each to each; and have likewise the angles contained by those sides equal to one another; they shall likewise have their bases, or third sides, equal; and the two triangles shall be equal; and their other angles shall be equal, each to each, viz. those to which the equal sides are opposite.

Prop. V. Theor. The angles at the base of an isosceles triangle are equal to one another; and, if the equal sides be produced, the angles upon the other side of the base shall be equal.

Prop. VI. Theor. If two angles of a triangle be

●qual to one another, the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another.

Prop. VII. Theor. Upon the same base, and on the same side of it, there cannot be two triangles that have their sides which are terminated in one extremity of the base equal to one another, and likewise those which are terminated in the other extremity.

Prop. VIII. Theor. If two triangles have two sides of the one equal to two sides of the other, each to each, and have likewise their bases equal; the angle which is contained by the two sides of the one shall be equal to the angle contained by the two sides equal to them, of the other.

Prop. IX. Prob. To bisect a given rectilineal angle, that is, to divide it into two equal angles. Prop. X. Prob. To bisect a given finite straight line, that is, to divide it into two equal parts.

Prop. XI. Prob. To draw a straight line at right angles to a given straight line, from a given point in the same.

Prop. XII. Prob. To draw a straight line perpendicular to a given straight line of an unlimited length, from a given point without it.

Prop. XIII. Theor. The angles which one straight line makes with another upon the one side of it, are either two right angles, or are together equal to two right augles.

Prop. XIV. Theor. If, at a point in a straight line, two other straight lines, upon the opposite sides of it, make the adjacent angles together equal to two right angles, these two straight lines shall be in one and the same straight line.

Prop. XV. Theor. If two straight lines cut one another, the vertical, or opposite, angles shall be equal.

Prop. XVI. Theor. If one side of a triangle be produced, the exterior angle is greater than either of the interior opposite angles.

Prop. XVII. Theor. Any two angles of a triangle are together less than two right angles. Prop. XVIII. Theor. The greater side of every triangle is opposite to the greater angle.

Prop. XIX. Theor. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it..

Prop. XX. Theor. Any two sides of a triangle are together greater than the third side.

Prop. XXI. Theor. If, from the ends of the side of a triangle, there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.

Prop. XXII. Prob. To make a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third.

Prop. XXIII. Prob. At a given point in a given straight line, to make a rectilineal angle equal to a given rectilineal angle.

Prop. XXIV. Theor. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them, of the other; the base of that which has the greater angle shall be greater than the base of the other.

Prop. XXV. Theor. If two triangles have two sides of the one equal to two sides of the other, each to each, but the base of the one greater than the base of the other; the angle also contained by the sides of that which has the greater base, shall VOL. V.

be greater than the angle contained by the sides equal to them, of the other.

Prop. XXVI. Theor. If two triangles have two angles of 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 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.

Prop. XXVII. Theor. If a straight line falling upon two other straight lines makes the alternate angles equal to one another, these two straight lines shall be parallel.

Prop. XXVIII. Theor. If a straight line falling upon two others traight lines makes the exterior angle equal to the interior and opposite upon the saine side of the line; or makes the interior angles upon the same side together equal to two right angles; the two straight lines shall be parallel to one another.

Prop. XXIX. Theor. 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 upon the same side; and likewise the two interior angles upon the same side together equal to two right angles.

Prop. XXX. Theor. Straight lines which are parallel to the same straight line are parallel to one another.

Prop. XXXI. Prob. To draw a straight line through a given point parallel to a given straight line.

Prop. XXXII. Theor. If a side of any triangle be produced, the exterior angle is equal to the two interior and opposite angles: and the three interior angles of every triangle are equal to two right angles.

Prop. XXXIII. Theor. The straight lines which join the extremities of two equal and parallel straight lines, towards the same parts, are also themselves equal and parallel.

Prop. XXXIV. Theor. The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them, that is, divides them in two equal parts.

Prop. XXXV. Theor. Parallelograms upon the saine base and between the same parallels, are equal to one another.

Prop. XXXVI. Theor. Parallelograms upon equal bases, and between the same parallels, are equal to one another.

Prop. XXXVII. Theor. Triangles upon the same base, and between the same parallels, are equal to one another.

Prop. XXXVIII. Theor. Triangles upon equal bases, and between the same parallels, are equal to one another.

Prop. XXXIX. Theor. Equal triangles upon the same base, and upon the same side of it, are between the same parallels.

Prop. XL. Theor. Equal triangles upon equal bases, in the same straight line, and towards the same parts, are between the same parallels.

Prop. XLI. Theor. If a parallelogram and triangle be upon the same base, and between the same parallels; the parallelogram shall be double of the triangle.

Prop. XLII. Prob. To describe a parallelogram that shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

S

Prop. XLIII. Theor. The complements of the parallelograms which are about the diameter of any parallelogram, are equal to one another.

Prop. XLIV. Prob. To a given straight line to apply a parallelogram, which shall be equal to a given triangle, and have one of its angles equal to a given rectilineal angle.

Prop. XLV. Prob. To describe a parallelogram equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle. Prop. XLVI. Prob. To describe a square upon a given straight line.

Prop. XLVII. Theor. In any right angled triangle, the square which is described upon the side subtending the right angle, is equal to the squares described upon the sides which contain the right angle,

Prop. XLVIII. Theor. 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. Def. 1.-Every right angled parallelogram is said to be contained by any two of the straight lines which contain one of the right angles. 2. In every parallelogram, any of the parallelograms about a diameter, together with the two complements, is called a gnomon.

Prop. 1. Theor. 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.

Prop. II. Theor. 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.

Prop. III. Theor. 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.

Prop. IV. Theor. 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.

Prop. V. Theor. 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.

Prop. VI. Theor. 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.

Prop. VII. Theor. 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.

Prop. VIII. Theor. If a straight line be divided into any two parts, four times the rectangle contained by the whole line, and one of the parts, to gether with the square of the other part, is equal to the square of the straight line which is made up of the whole and that part.

Prop. IX. Theor. 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.

Prop. X. Theor. 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.

Prop. XI. Prob. To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.

a

Prop. XII. Theor. In obtuse angled triangles, if 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 ob tuse angle, by twice the rectangle contained by the side upon which, when produced, the perpendicular falls, and the straight line intercepted without the triangle between the perpendicular and the obtuse angle.

Prop. XIII. Theor. 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 perpendicular let fall upon it from the opposite angle, and the acute angle.

Prop. XIV. Prob. To describe a square that shall be equal to a given rectilineal figure.

Book III. Def. 1.-Equal circles are those of which the diameters are equal, or from the centres of which the straight lines to the circumferences are equal.

This is not a definition but a theorem, the truth of which is evident; for, if the circles be applied to one another, so that their centres coincide, the circles must likewise coincide, since the straight lines from the centres are equal.

2. A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it.

3. Circles are said to touch one another, which meet, but do not cut one another.

4. Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal.

5. And the straight line on which the greater perpendicular falls, is said to be farther from the

centre.

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

7. The angle of a segment is that which is contained by the straight line and the circumfer

ence.

8. An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line which is the base of the seg ment.

9. And an angle is said to insist or stand upon the circumference intercepted between the straight lines that contain the angle.

10. The sector of a circle is the figure contain. ed by two straight lines drawn from the centre, and the circumference between them.

11. Similar segments of a circle, are those in which the angles are equal, or which contain equal angles.

cle.

Prop. I. Prob. To find the centre of a given cir

Prop. II. Theor. If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle.

Prop. III. Theor. If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it shall cut it at right angles; and, if it cuts it at right angles, it shall bisect it.

Prop. IV. Theor. If in a circle two straight lines cut one another which do not both pass through the centre, they do not bisect each other.

Prop. V. Theor. If two circles cut one another, they shall not have the same centre.

Prop. VI. Theor. If two circles touch one another internally, they shall not have the same

centre.

Prop. VII. Theor. 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 circumference, 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 which passes through the centre is always greater than one more remote: 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.

Prop. VIII. Theor. 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.

Prop. IX. Prob. If a point be taken within a circle, from which there fall more than two equal straight lines to the circumference, that point is the centre of the circle.

Prop. X. Theor. One circumference of a circle cannot cut another in more than two points.

Prop. XI. Theor. If two circles touch each other internally, the straight line which joins their centres being produced shall pass through the point of contact.

Prop. XII. Theor. If two circles touch each other externally, the straight line which joins their centres shall pass through the point of contact.

Prop. XIII. Theor. One circle cannot touch another in more points than one, whether it touches it on the inside or outside.

Prop. XIV. Theor. 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.

Prop. XV. Theor. The diameter is the greatest straight line in a circle; and, of all others, that which is nearer to the centre is always greater than one more remote; and the greater is nearer to the centre than the less.

Prop. XVI. Theor. 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, so as not to cut the circle; or, which is the same thing, no straight line can make so great an acute

angle with the diameter at its extremity, or su small an angle with the straight line which is at right angles to it, as not to cut the circle.

Prop. XVII. Prob. To draw a straight line from a given point, either without or in the circumference, which shall touch a given circle.

Prop. XVIII. Theor. If a straight line touches a circle, the straight line drawn from the centre to the point of contact, shall be perpendicular to the line touching the circle.

Prop. XIX. Theor. If a straight line touches a circle, and from the point of contact a straight line be drawn at right angles to the touching line, the centre of the circle shall be in that line.

Prop. XX. Theor. The angle at the centre of a circle is double of the angle at the circumference, upon the same base, that is, upon the same part of the circumference.

Prop. XXI. Theor. The angles in the same segment of a circle are equal to one another.

Prop. XXII. Theor. The opposite angles of any quadrilateral figure described in a circle, are together equal to two right angles.

Prop. XXIII. Theor. Upon the same straight line, and upon the same side of it, there cannot be two similar segments of circles, not coinciding with one another.

Prop. XXIV. Theor. Similar segments of circles upon equal straight lines, are equal to one another.

Prop. XXV. Prob. A segment of a circle being given, to describe the circle of which it is the segment.

Prop. XXVI. Theor. In equal circles, equal angles stand upon equal circumferences, whether they be at the centres or circumferences.

Prob. XXVII. Theor. In equal circles, the angles which stand upon equal circumferences are equal to one another, whether they be at the centres or circumferences.

Prop. XXVIII. Theor. In equal circles, equal straight lines cut off equal circumferences, the greater equal to the greater, and the less to the less.

Prop. XXIX. Theor. In equal circles equal circumferences are subtended by equal straight lines.

Prop. XXX. Prob. To bisect a given circumfer ence, that is, to divide it into two equal parts.

Prop. XXXI. Theor. In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle.

Prop. XXXII. Theor. If a straight line touches a circle, and from the point of contact a straight line be drawn cutting the circle, the angles made by this line with the line touching the circle, shall be equal to angles which are in the alternate segments of the circle.

Prop. XXXIII. Prob. Upon a given straight line to describe a segment of a circle, containing an angle equal to a given rectilineal angle.

Prop. XXXIV. Prob. To cut off a segment from a given circle which shall contain an angle equal to a given rectilineal angle.

Prop. XXXV. Theor. If two straight lines within a circle cut one another, the rectangle contained> by the segments of one of them is equal to the rectangle contained by the segments of the other.

Prop. XXXVI. Theor. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it; the rectangle contained by the whole line which

cuts the circle, and the part of it without the cirele, shall be equal to the square of the line which touches it.

Pop. XXXVII. Theor. If from a point without a circle there be drawn two straight lines, one of which cuts the circle, and the other meets it; if the rectangle contained by the whole line which cuts the circle, and the part of it without the eircle be equal to the square of the line which meets it, the line which meets shall touch the circle.

Book IV. Def. 1.-A rectilinea! figure is said to be inscribed in another rectilincal figure, when all the angles of the inscribed figure are upon the side of the figure in which it is inscribed, each upon each. 2. In like manner, a figure is said to be describe ed about another figure, w en all the sides of the circumscribed figure pass through the angular points of the figure about which it is described, each through each.

3. A rectilineal figure is said to be inscribed in a circle, when all the angles of the inscribed figure are upon the circumference of the circle.

4. A rectilineal figure is said to be described about a circle, when each side of the circumscribed figure touches the circumference of the circle. 5. In like manner, a circle is said to be inscribed in a rectilineal figure, when the circumference of the circle touches each side of the figure.

6. A circle is said to be described about a rectilineal figure, when the circumference of the circle passes through all the angular points of the figure about which it is described.

7. A straight line is said to be placed in a circle, when the extremities of it are in the circumference of the circle.

Prop. I. Prob. In a given circle to place a straight line, equal to a given straight line not greater than the diameter of the circle.

Prop. II. Prob. In a given circle to inscribe a triangle equiangular to a given triangle.

Prop. III. Prob. About a given circle to describe a triangle equiangular to a given triangle. Prop. IV. Prob. To inscribe a circle in a given triangle.

Prop. V. Prob. To describe a circle about a given triangle.

Prop. VI. Prob. To inscribe a square in a given circle.

Prop. VII. Prob. To describe a square about a given circle.

Prop. VIII. Prob. To inscribe a circle in a given square.

Prop. IX. Prob. To describe a circle about a given square.

Prop. X. Prob. To describe an isosceles triangle, having each of the angles at the base double of the third angle.

Prop. XI. Prob. To inscribe an equilateral and equiangular pentagon in a given circle.

Prop. XII. Prob. To describe an equilateral and equiangular pentagon about a given circle.

Prop. XIII. Prob. To inscribe a circle in a given equilateral and equiangular pentagon.

Prop. XIV. Prob. To describe a circle about a given equilateral and equiangular pentagon. Prop. XV. Prob. To inscribe an equilateral and equiangular hexagon in a given circle.

Prop. XVI. Prob. To inscribe an equilateral and equiangular quindecagon in a given circle.

Book V. Def. 1.-A less magnitude is said to be a part of a greater magnitude, when the less measures the greater, that is, when the less is contained a certain number of times exactly in the greater.

2. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, when the greater contains the less a certain number of times exactly.

3. Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity.

4. Magnitudes are said to have a ratio to one another, when the less can be multiplied so as to exceed the other.

5. The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth; if the multiple of the first be less than that of the second, the multiple of the third is also less than that of the fourth; or, if the multiple of the first be equal to that of the second, the multiple of the third is also equal to that of the fourth; or, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth.

6. Magnitudes which have the same ratio are called proportionals. N. B. When four magnitudes are proportionals, it is usually expressed by saying, the first is to the second, as the third to the fourth.

7. When of the equimultiples of four magnitudes (taken as in the fifth definition) the multiple of the first is greater than that of the second, but the multiple of the third is not greater than the multiple of the fourth; then the first is said to have to the second a greater ratio than the third magnitude has to the fourth; and, on the contrary, the third is said to have to the fourth a less ratio than the first has to the second.

8. Analogy, or proportion, is the similitude

of ratios.

9. Proportion consists in three terms at least. 10. When three magnitudes are proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second.

11. When four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second, and so on, quadruplicate, &c. increasing the denomination still by unity, in any number of proportionals.

Definition A. to wit, of compound ratio. When there are any number of magnitudes, of the same kind, the first is said to have to the last of them the ratio compounded of the ratio which the first has to the second, and of the ratio which the second has to the third, and of the ratio which the third has to the fourth, and so on unto the last magnitude.

For example, if A, B, C, D be four magnitudes of the same kind, the first A is said to have to the last D the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D; or, the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D: and if A has to B the same ratio which E has to F; and B to C, the same ratio that G has to H; and C to D, the same that K has to L; then, by this definition, A is said to have to D the ratio compounded of ratios which are the same with the ratios of E to F, G to H, and K to L: and the same thing is to be understood when it is more briefly expressed, by saying A has to D the ratio compounded of the ratios of E to F, G to H, and K to L.

In like manner, the same things being supposed