The (almost really) Complete Works of Lewis Carroll

45. Under what circumstances does Euclid assert, as an Axiom, that two Lines will meet?

62. Define “A fortiori.” Give an instance.
63. Define “Reductio ad absurdum.” Give an instance.
64. When is a Line said to “subtend” an angle?

1. Probl. To describe an equilateral Triangle on a given finite straight Line.
2. Probl. From a given Point to drawa straight Line equal to a given straight Line.
3. Probl. From the greater of two given straight Lines to cut off a part equal to the less.
4. Theor. If two Triangles have two sides of the one equal to two sides of the other, each to each, and the angles included by those sides equal; the bases or third sides are equal; as also are the Triangles; and their other angles are equal, viz. those to which the equal sides are opposite.
5. Theor. The angles at the base of an isosceles Triangle are equal; and, if the equal sides be produced, the angles on the other side of the base are equal.
Cor. Every equilateral Triangle is also equiangular.
6. Theor. If two angles of a Triangle be equal; the sides also, which subtend, or are opposite to, the equal angles, are equal.
Cor. Every equiangular Triangle is also equilateral.
7. Theor. On the same base, and on the same side of it, there cannot be two Triangles which have their sides terminated in one extremity of the base equal, and at the same time their sides terminated in the other extremity equal.
8. Theor. If two Triangles have two sides of the one equal to two sides of the other, each to each, and likewise their bases equal; the angle contained by the two sides of the one is equal to the angle contained by the two sides, equal to them, of the other.
9. Probl. To bisect a given rectilineal angle.
10. Probl. To bisect a given finite straight Line.
11. Probl. To draw a straight Line at right angles to a given straight Line from a given Point in the same.
12. Probl. To draw a straight Line perpendicular to a given straight Line of unlimited length, from a given Point without it.
13. Theor. The angles, which one straight Line makes with another on one side of it, are either two right angles, or are together equal to two right angles.
Cor. 1. The angles at the point of intersection of two straight Lines are together equal to four right angles.
Cor. 2. All the angles, made by any number of Lines meeting in one Point, are together equal to four right angles.
14. Theor. If, at a Point in a straight Line, two other straight Lines on opposite sides of it make the adjacent angles together equal to two right angles; these two straight Lines are in one and the same straight Line.
15. Theor. If two straight Lines cut one another; the vertical, or opposite, angles are equal.
16. Theor. If one side of a Triangle be produced; the exterior angle is greater than either of the interior opposite angles.
17. Theor. Any two angles of a Triangle are together less than two right angles.
18. Theor. The greater side of every Triangle is opposite to the greater angle.
19. Theor. The greater angle of every Triangle is subtended by the greater side, or has the greater side opposite to it.
20. Theor. Any two sides of a Triangle are together greater than the third.
21. Theor. If from the ends of one side of a Triangle two straight Lines be drawn to a Point within the Triangle; these two straight Lines are less than the other two sides of the Triangle, but contain a greater angle.
22. Probl. To make a Triangle, the sides of which shall be equal to three given straight Lines, but any two whatever of these must be greater than the third.
23. Probl. At a given Point in a given straight Line to make an angle equal to a given rectilineal angle.
24. 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 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 is greater than the base of the other.
25. 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, contained by the sides of that which has the greater base, is greater than the angle contained by the sides, equal to them, of the other.
26. Theor. If two Triangles have 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 opposite to the equal angles in each: the other sides are equal, each to each, and also the third angle of the one to the third angle of the other.
27. Theor. If a straight Line falling upon two other straight Lines make the alternate angles equal; the two straight Lines are parallel.
28. Theor. If a straight Line falling upon two other straight Lines make the exterior angle equal to the interior and opposite angle on the same side of the Line, or make the two interior angles on the same side together equal to two right angles; the two straight Lines are parallel.
29. Theor. If a straight Line fall upon two parallel straight Lines; it makes the alternate angles equal, and the exterior angle equal to the interior and opposite upon the same side; and the two interior angles upon the same side together equal to two right angles.
30. Theor. Straight Lines which are parallel to the same straight Line are parallel to one another.
31. Probl. To draw a straight Line through a given Point parallel to a given straight Line.
32. 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.
Cor. 1. All the interior angles of any rectilineal Figure, together with four right angles, are equal to twice as many right angles as the Figure has sides.
Cor. 2. All the exterior angles of any rectilineal Figure are equal to four right angles.
33. Theor. The straight Lines which join the extremities of equal and parallel straight Lines towards the same parts are themselves equal and parallel.
34. Theor. The opposite sides and angles of a Parallelogram are equal; and the diagonal bisects it.
35. Theor. Parallelograms on the same base, and between the same parallels, are equal.
36. Theor. Parallelograms upon equal bases, and between the same parallels, are equal.
37. Theor. Triangles on the same base, and between the same parallels, are equal.
38. Theor. Triangles upon equal bases, and between the same parallels, are equal.
39. Theor. Equal Triangles upon the same base, and upon the same side of it, are between the same parallels.
40. Theor. Equal Triangles upon equal bases in the same straight Line, and towards the same parts, are between the same parallels.
41. Theor. If a Parallelogram and a Triangle be upon the same base, and between the same parallels; the Parallelogram is double of the Triangle.
42. Probl. To describe a Parallelogram equal to a given Triangle, and having an angle equal to a given rectilineal angle.
43. Theor. The complements of the Parallelograms, which are about the diagonal of any Parallelogram, are equal.
44. Probl. To a given straight Line to apply a Parallelogram equal to a given Triangle, and having an angle equal to a given rectilineal angle.
45. Probl. To describe a Parallelogram equal to a given rectilineal Figure, and having an angle equal to a given rectilineal angle.
Cor. To a given straight Line to apply a Parallelogram equal to a given rectilineal Figure, and having an angle equal to a given rectilineal angle.
46. Probl. To describe a Square upon a given straight Line.
Cor. A Parallelogram, that has one right angle, is rectangular.
47. Theor. In a right-angled Triangle, the Square, described upon the side subtending the right angle, is equal to the Squares described upon the sides containing it.
48. Theor. If the Square described upon one side of a Triangle be equal to the Squares described upon the other two; the angle contained by those sides is a right angle.

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.
2. 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.
3. 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 aforesaid part.
4. 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.
Cor. The Parallelograms about the diagonal of a Square are likewise Squares.
5. Theor. If a straight Line be divided into two equal, 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. The difference of the squares of two un-unequal Lines is equal to the rectangle contained by their sum and difference.
6. Theor. If a straight Line be bisected, and produced to any Point; the rectangle contained by the whole Line produced, and the part of it produced, together with the square of half the Line bisected, is equal to the square of the Line made up of the half and the part produced.
7. 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.
8. 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, together with the square of the other part, is equal to the square of the Line made up of the whole and that part.
9. Theor. If a straight Line be divided into two equal, and also into two unequal parts; the squares of the unequal parts are together double of the square of half the Line, and of the square of the Line between the points of section.
10. 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, and of the square of the Line made up of the half and the part produced.
11. Probl. 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.
12. Theor. In an obtuse-angled Triangle, if a perpendicular be drawn from either 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 it 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.
13. 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.
14. Probl. To describe a Square equal to a given rectilineal Figure.

1. When are Circles said to be equal?
2. When is a straight Line said to touch a Circle?
3. When are Circles said to touch each other?
4. When are straight Lines said to be equally distant from the centre of a Circle?
5. When is one of two straight Lines said to be further than the other from the centre of a Circle?
6. Define an arc of a Circle.
7. Define “a segment of a Circle.”
8. Define “the angle of a segment.”
9. Define “an angle in a segment.”
10. When is an angle said to “stand upon” part of the circumference of a Circle (i. e. upon an arc)?
11. When is part of the circumference of a Circle (i. e. an arc) said to be “subtended” by an angle?
12. Define “a sector of a Circle.”
13. Define “similar segments of Circles.”
14. When is a segment of a Circle said to be “alternate” with an angle? Illustrate your answer by a figure.

1. Probl. To find the centre of a given Circle.
Cor. If in a Circle a straight Line bisect another at right angles, the centre of the Circle is on the Line which bisects the other.
2. Theor. If any two Points be taken in the circumference of a Circle, the straight Line which joins them falls within the Circle.
3. 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 cuts it at right angles; and if it cut it at right angles, it bisects it.
4. Theor. If in a Circle two straight Lines cut each other, which do not both pass through the centre, they do not bisect each other.
5. Theor. If two Circles cut each other, they have not the same centre.
6. Theor. If one Circle touch another internally, they have not the same centre.
7. Theor. Of all the straight Lines drawn from a Point in a Circle, not the centre, to the circumference, the greatest is that which passes through the centre; and the other part of that diameter is the least; and of the others, that which is the nearer to the one through the centre is greater than the more remote; and only two equal straight Lines can be drawn from the Point to the circumference, one on each side of the least.
8. Theor. If from a Point without a Circle, straight Lines be drawn to the circumference; of those which fall on the concave circumference, the greatest is that which passes through the centre; and of the rest, that which is nearer to the greatest is greater than the more remote; but of those which fall on 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 less than the more remote; and only two equal straight Lines can be drawn from the Point to the circumference, one on each side of the least.
9. Theor. If, from a Point within a Circle, more than two equal straight Lines be drawn to the circumference, that Point is the centre of the Circle.
10. Theor. One Circle cannot cut another in more than two Points.
11. Theor. If one Circle touch another internally, the straight Line which joins their centres, being produced, passes through the Point of contact.
12. Theor. If two Circles touch each other externally, the straight Line which joins their centres passes through the Point of contact.
13. Theor. One Circle cannot touch another in more Points than one, whether it touch internally or externally.
14. Theor. Equal straight Lines in a Circle are equally distant from the centre; and those which are equally distant from the centre are equal.
15. Theor. The diameter is the greatest straight Line in a Circle; and of all others, that which is nearer to the centre is greater than the more remote; and the greater is nearer to the centre than the less.
16. 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 so small an angle with the straight Line which is at right angles to it, as not to cut the circle.
Cor. The straight Line which is drawn at right angles to the diameter of a Circle from the extremity of it, touches the Circle; and it touches it only in one Point. Also, there cannot be more than one straight Line touching the Circle in the same Point.
17. Probl. To draw a straight Line from a given Point, either without or in the circumference, which shall touch a given Circle.
18. Theor. If a straight Line touch a Circle, the straight Line drawn from the centre to the Point of contact is at right angles to the Line touching the Circle.
19. Theor. If a straight Line touch 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 is in that Line.
20. 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.
21. Theor. The angles in the same segment of a Circle are equal.
22. Theor. The opposite angles of any quadrilateral Figure inscribed in a Circle are together equal to two right angles.
23. Theor. On the same straight Line, and on the same side of it, there cannot be two similar
segments of Circles not coinciding with each other.
24. Theor. Similar segments of Circles upon equal straight Lines are equal.
25. Probl. A segment of a Circle being given, to describe the Circle, of which it is the segment.
26. Theor. In equal Circles, equal angles stand upon equal circumferences, whether they be at the centres or circumferences.
27. Theor. In equal Circles, the angles which stand upon equal circumferences are equal, whether they be at the centres or circumferences.
28. Theor. In equal Circles, equal straight Lines cut off equal circumferences, the greater equal to the greater, and the less to the less.
29. Theor. In equal Circles, equal circumferences are subtended by equal straight Lines.
30. Probl. To bisect a given circumference, or arc.
31. 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.
Cor. If one angle of a Triangle be equal to the other two, it is a right angle.
32. Theor. If a straight Line touch 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 are equal to the angles which are in the alternate segments of the Circle.
33. Probl. Upon a given straight Line to describe a segment of a Circle containing an angle equal to a given rectilineal angle.
34. Probl. To cut off a segment from a given Circle, which shall contain an angle equal to a given rectilineal angle.
35. Theor. If two straight Lines in a Circle cut each other, the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other.
36. 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 Circle, is equal to the square of the Line which touches it.
Cor. If from a Point without a Circle there be drawn two straight Lines cutting it, the rectangles contained by the whole Lines and the parts of them without the Circle, are equal.
37. 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 Circle, be equal to the square of the Line which meets it, the Line which meets the Circle touches it.

7. When is a straight Line said to be placed in a Circle?

1. Probl. In a given Circle, to place a straight Line equal to a given straight Line, which is not greater than the diameter of the Circle.
2. Probl. In a given Circle to inscribe a Triangle equiangular to a given Triangle.
3. Probl. About a given Circle to describe Triangle equiangular to a given Triangle.
4. Probl. To inscribe a Circle in a given Triangle.
5. Probl. To describe a Circle about a given Triangle.
Cor. If the centre of the Circle be within the Triangle, each of its angles is less than a right angle; but, if the centre be in one of the sides of the Triangle, the angle opposite to this side is a right angle; and, if the centre be without the Triangle, the angle, opposite to the side beyond which it is, is greater than a right angle. Also, if the given Triangle be acute-angled, the centre of the Circle is within it; if it be right-angled, the centre is in the side opposite to the right angle; and if it be obtuse-angled, the centre is without the Triangle, beyond the side opposite to the obtuse angle.
6. Probl. To inscribe a Square in a given Circle.
7. Probl. To describe a Square about a given Circle.
8. Probl. To inscribe a Circle in a given Square.
9. Probl. To describe a Circle about a given Square.
10. Probl. To describe an isosceles Triangle, having each of the angles at the base double of the third angle.
11. Probl. To describe an equilateral and equiangular Pentagon in a given Circle.
12. Probl. To describe an equilateral and equiangular Pentagon about a given Circle.
13. Probl. To inscribe a Circle in a given equilateral and equiangular Pentagon.
14. Probl. To describe a Circle about a given equilateral and equiangular Pentagon.
15. Probl. To inscribe an equilateral and equiangular Hexagon in a given Circle.
Cor. The side of the Hexagon is equal to the straight Line from the centre, that is, to the semidiameter of the Circle.
16. Probl. To inscribe an equilateral and equiangular Quindecagon in a given Circle.

31. What condition must be fulfilled by two magnitudes, in order that each of them may be said to have a ratio to the other?
32. Why has Euclid laid down this condition?
33. What is the least number of terms which can be proportionals?
34. Show the identity of the two forms of “dividendo.”

1. Theor. If any number of magnitudes be equimultiples of as many, each of each; what multiple soever any one of them is of its part, the same multiple are all the first magnitudes of all the others.
2. Theor. If the first magnitude be the same multiple of the second that the third is of the fourth, and the fifth the same multiple of the second that the sixth is of the fourth; then the first and fifth together are the same multiple of the second that the third and sixth together are of the fourth.
Cor. If any number of magnitudes A, B, C, &c., be multiples of another X; and as many a, b, c, &c., be the same multiples of x, each of each; the whole of the first, viz. $A+B+C+\text{\&c.}$, is the same multiple of X, that the whole of the last, viz. $a+b+c+\text{\&c.}$, is of x.
3. Theor. If the first be the same multiple of the second which the third is of the fourth; and if of the first and third there be taken equimultiples, these are equimultiples, the one of the second, and the other of the fourth.
4. Theor. If the first of four magnitudes have the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third have the same ratio to any equimultiples of the second and fourth, viz.: ‘the equimultiple of the first has the same ratio to that of the second, which the equimultiple of the third has to that of the fourth.’
Cor. Likewise, if the first have the same ratio to the second, which the third has to the fourth, then also, any equimultiples whatever of the first and third have the same ratio to the second and fourth: And in like manner, the first and the third have the same ratio to any equimultiples whatever of the second and fourth.
5. Theor. If one magnitude be the same multiple of another, which a magnitude taken from the first is of a magnitude taken from the other; the remainder is the same multiple of the remainder that the whole is of the whole.
6. Theor. If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two; the remainders are either equal to these others, or equimultiples of them.
A. Theor. If the first of four magnitudes have to the second the same ratio which the third has to
the fourth; then, if the first be greater than the second, the third is also greater than the fourth; if equal, equal; and if less, less.
B. Theor. If four magnitudes be proportionals, they are proportionals also when taken inversely.
C. Theor. If the first be the same multiple or part of the second, that the third is of the fourth; the first is to the second, as the third to the fourth.
D. Theor. If the first be to the second, as the third to the fourth; and if the first be a multiple or part of the second; the third is the same multiple or part of the fourth.
7. Theor. Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes.
8. Theor. Of unequal magnitudes, the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less, than it has to the greater.
9. Theor. Magnitudes which have the same ratio to the same magnitude, are equal; and those to which the same magnitude has the same ratio, are equal.
10. Theor. That magnitude which has a greater ratio than another has to the same magnitude, is the greater; and that magnitude to which the same has a greater ratio than it has to another magnitude, is the lesser.
11. Theor. Ratios that are the same to the same ratio, are the same to one another.
12. Theor. If any number of magnitudes be proportionals, as one of the antecedents is to its consequent, so are all the antecedents taken together to all the consequents.
13. Theor. If the first have to the second the same ratio which the third has to the fourth, but the third to the fourth a greater ratio than the fifth to the sixth; the first has to the second a greater ratio than the fifth to the sixth.
Cor. And if the first have to the second a greater ratio than the third has to the fourth, but the third the same ratio to the fourth which the fifth has to the sixth; the first has to the second a greater ratio than the fifth has to the sixth.
14. Theor. If the first have to the second the same ratio which the third has to the fourth: then if the first be greater than the third, the second is greater than the fourth, if equal, equal; and if less, less.
15. Theor. Magnitudes have the same ratio to each other which their equimultiples have.
16. Theor. If four magnitudes of the same kind be proportionals, they are also proportionals when taken alternately.
17. Theor. If magnitudes, taken jointly, be proportionals, they are also proportionals when taken separately; that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two has to the other the same ratio which the remaining one of the last two has to the other of these.
*17. Theor. (Stated according to the Definition of “Dividendo.”) If the first of four magnitudes have to the second the same ratio which the third has to the fourth; the excess of the first above the second has to the second the same ratio which the excess of the third above the fourth has to the fourth.
18. Theor. If magnitudes, taken separately, be proportionals, they are also proportionals when taken jointly; that is, if the first be to the second, as the third to the fourth, the first and second together are to the second as the third and fourth together to the fourth.
19. Theor. If a whole magnitude be to a whole as a magnitude taken from the first is to a magnitude taken from the other, the remainder is to the remainder as the whole to the whole.
Cor. If the whole be to the whole, as a magnitude taken from the first is to a magnitude taken from the other; the remainder likewise is to the remainder as the magnitude taken from the first to that taken from the other.
E. Theor. If four magnitudes be proportionals, they are also proportionals by conversion; that is, the first is to its excess above the second, as the third to its excess above the fourth.
20. Theor. If there be three magnitudes, and other three, which taken two and two, have the same ratio; if the first be greater than the third, the fourth is greater than the sixth; if equal, equal; and, if less, less.
21. Theor. If there be three magnitudes, and other three, which have the same ratio, taken two and two, but in a cross order; if the first magnitude be greater than the third, the fourth is greater than the sixth; if equal, equal; and, if less, less.
22. Theor. If there be any number of magnitudes, and as many others, which, taken two and two in order, have the same ratio; the first has to the last of the first magnitudes the same ratio which the first of the others has to the last.
23. Theor. If there be any number of magnitudes, and as many others, which, taken two and two in a cross order, have the same ratio; the first has to the last of the first magnitudes the same ratio which the first of the others has to the last.
24. Theor. If the first have to the second the same ratio which the third has to the fourth, and the fifth to the second the same ratio which the sixth has to the fourth; the first and fifth together have to the second the same ratio which the third and sixth together have to the fourth.
Cor. 1. If the same hypothesis be made, the excess of the first and fifth is to the second as the excess of the third and sixth to the fourth.
Cor. 2. The Proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the same ratio that the corresponding one of the second rank has to a fourth magnitude.
25. Theor. If four magnitudes of the same kind be proportionals, the greatest and least of them together are greater than the other two together.
F. Theor. Ratios which are compounded of the same ratios, are the same to one another.
G. Theor. If several ratios be the same to several ratios, each to each; the ratio which is compounded of ratios which are the same to the first ratios, each to each, shall be the same to the ratio compounded of ratios which are the same to the other ratios, each to each.
H. Theor. If a ratio which is compounded of several ratios be the same to a ratio which is compounded of several other ratios; and if one of the first ratios, or the ratio which is compounded of several of them, be the same to one of the last ratios, or to the ratio which is compounded of several of them; then the remaining ratio of the first, or, if there be more than one, the ratio compounded of the remaining ratios, shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio compounded of these remaining ratios.
K. Theor. If there be any number of ratios, and any number of other ratios such, that the ratio which is compounded of ratios which are the same to the first ratios, each to each, is the same to the ratio which is compounded of ratios which are the same, each to each, to the last ratios; and if one of the first ratios, or the ratio which is compounded of ratios which are the same to several of the first ratios, each to each, be the same to one of the last ratios, or to the ratio which is compounded of ratios which are the same, each to each, to several of the last ratios; then the remaining ratio of the first, or, if there be more than one, the ratio which is compounded of ratios which are the same, each to each, to the remaining ratios of the first shall be the same to the remaining ratio of the last, or, if there be more than one, to the ratio which is compounded of ratios which are the same, each to each, to these remaining ratios.

1. Define the “altitude” of a Triangle or Parallelogram.
2. When are rectilineal Figures said to be similar?
3. When are two Triangles said to have two sides of the one “reciprocally proportional” to two sides of the other?
4. When is a straight Line said to be cut in extreme and mean ratio?
5. When are two Lines said to be similarly divided?

1. Theor. Triangles and Parallelograms of the same altitude are one to another as their bases.
Cor. Triangles and Parallelograms of equal altitudes are one to another as their bases.
2. Theor. If a straight Line be drawn parallel to one of the sides of a Triangle, it cuts the other sides, or those produced, proportionally; and if two sides, or two sides produced, be cut proportionally, the straight Line which joins the Points of section is parallel to the remaining side.
3. Theor. If the vertical angle of a Triangle be bisected by a straight Line cutting the base, the segments of the base have the same ratio as the other sides of the Triangle; and if the segments of the base have the same ratio as the other sides of the Triangle, the straight Line, drawn from the vertex to the Point of section, bisects the vertical angle.
A. Theor. If the exterior angle of a Triangle, made by producing one of its sides, be bisected by a straight Line, which also cuts the base produced; the segments between the dividing Line and the extremities of the base have the same ratio as the other sides of the Triangle; and if the segments of the base produced have the same ratio as the other sides of the Triangle, the straight Line, drawn from the vertex to the Point of section, bisects the exterior angle of the Triangle.
4. Theor. The sides about the equal angles of equiangular Triangles are proportionals; and those which are opposite to the equal angles are homologous.
5. Theor. If the sides of two Triangles about each of their angles be proportionals, the Triangles are equiangular, and have those angles equal which are opposite to the homologous sides.
6. Theor. If two Triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals; the Triangles are equiangular, and have those angles equal which are opposite to the homologous sides.
7. Theor. If two Triangles have one angle of the one equal to one angle of the other, and the sides about two other angles proportionals; then, if each of the remaining angles be either less, or not less, than a right angle, or if one of them be a right angle, the Triangles are equiangular, and have those angles equal about which the sides are proportionals.
8. Theor. In a right-angled Triangle, if a perpendicular be drawn from the right angle to the base, the Triangles on each side of it are similar to the whole Triangle, and to one another.
Cor. The perpendicular, drawn from the right angle of a right-angled Triangle to the base, is a mean proportional between the segments of the base. Also, each of the sides is a mean proportional between the base and its segment adjacent to that side.
9. Probl. From a given straight Line to cut off any part required.
10. Probl. To divide a given straight Line similarly to a given divided straight Line.
11. Probl. To find a third proportional to two given straight Lines.
12. Probl. To find a fourth proportional to three given straight Lines.
13. Probl. To find a mean proportional between two given straight Lines.
14. Theor. Equal Parallelograms which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and Parallelograms, which have an angle of the one equal to an angle of the other, and their sides about the equal angles reciprocally proportional, are equal.
15. Theor. Equal Triangles, which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional; and Triangles, which have an angle of the one equal to an angle of the other, and their sides about the equal angles reciprocally proportional, are equal.
16. Theor. If four straight Lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means; and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight Lines are proportionals.
17. Theor. If three straight Lines be proportionals, the rectangle contained by the extremes is equal to the square of the mean; and if the rectangle contained by the extremes be equal to the square of the mean, the three straight Lines are proportionals.
18. Probl. Upon a given finite straight Line to describe a Rectilineal Figure similar and similarly situated to a given rectilineal Figure.
19. Theor. Similar Triangles have to one another the duplicate ratio of their homologous sides.
Cor. If three straight Lines be proportionals, as the first is to the third, so is any Triangle upon the first to a similar and similarly described Triangle upon the second.
20. Theor. Similar Polygons may be divided into the same number of similar Triangles, having to one another the same ratio as the Polygons; and the Polygons have to one another the duplicate ratio of their homologous sides.
Cor. 1. Similar rectilineal Figures have to one another the duplicate ratio of their homologous sides.
Cor. 2. If three straight Lines be proportionals, as the first is to the third, so is any rectilineal Figure upon the first, to a similar and similarly described rectilineal Figure upon the second.
21. Theor. Rectilineal Figures, which are similar to the same rectilineal Figures, are also similar to one another.
22. Theor. If four straight Lines be proportionals, the similar rectilineal Figures similarly described upon them are also proportionals; and if the similar rectilineal Figures similarly described upon four straight Lines be proportionals, those straight Lines are proportionals.
23. Theor. Equiangular Parallelograms have to one another the ratio which is compounded of the ratios of their sides.
24. Theor. The Parallelograms about the diameter of any Parallelogram are similar to the whole and to one another.
25. Probl. To describe a rectilineal Figure similar to one, and equal to another, given rectilineal Figure.
26. Theor. If two similar Parallelograms have a common angle and be similarly situated, they are about the same diameter.
30. Probl. To divide a given straight Line in extreme and mean ratio.
31. Theor. In right-angled Triangles, the rectilineal Figure described on the side opposite to the right angle, is equal to the similar and similarly described Figures on the sides containing it.
32. Theor. If two Triangles, which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel; the remaining sides are in a straight Line.
33. Theor. In equal Circles, angles, whether at the centres or circumferences, have the same ratio which the circumferences have to one another; so also have the sectors.
B. Theor. If the vertical angle of a Triangle be bisected by a straight Line, which likewise cuts the base; the rectangle contained by the sides of the Triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight Line bisecting the angle.
C. Theor. If from the vertical angle of a Triangle a straight Line be drawn perpendicular to the base; the rectangle contained by the sides of the Triangle is equal to the rectangle contained by the perpendicular and the diameter of the Circle described about the Triangle.
D. Theor. The rectangle, contained by the diagonals of a quadrilateral Figure inscribed in a Circle, is equal to the rectangles contained by the opposite sides.