The Euclid's Elements of Geometry

Geometry is the study of figures. Euclid's Elements provides themost fundamental way of learning geometry geometrically.

Book X: Classification of incommensurables, II

Definitions

1. Given a rational straight line and a binomial, divided into its terms, such that the square on the greater term is greater than the square on the lesser by the square on a straight line commensurable in length with the greater, then, if the greater term is commensurable in length with the rational straight line set out, let the whole be called a first binomial straight line;
2. But if the lesser term is commensurable in length with the rational straight line set out, let the whole be called a second binomial;
3. And if neither of the terms is commensurable in length with the rational straight line set out, let the whole be called a third binomial.
4. Again, if the square on the greater term is greater than the square on the lesser by the square on a straight line incommensurable in length with the greater, then, if the greater term is commensurable in length with the rational straight line set out, let the whole be called a fourth binomial;
5. If the lesser, a fifth binomial;
6. And, if neither, a sixth binomial.

Propositions

48.  To find the first binomial line.
49. To find the second binomial line.
50. To find the third binomial line.
51. To find the fourth binomial line.
52. To find the fifth binomial line.
53. To find the sixth binomial line.
54. If an area is contained by a rational straight line and the first binomial, then the side of the area is the irrational straight line which is called binomial.
55. If an area is contained by a rational straight line and the second binomial, then the side of the area is the irrational straight line which is called a first bimedial.
56. If an area is contained by a rational straight line and the third binomial, then the side of the area is the irrational straight line called a second bimedial.
57. If an area is contained by a rational straight line and the fourth binomial, then the side of the area is the irrational straight line called major.
58. If an area is contained by a rational straight line and the fifth binomial, then the side of the area is the irrational straight line called the side of a rational plus a medial area.
59. If an area is contained by a rational straight line and the sixth binomial, then the side of the area is the irrational straight line called the side of the sum of two medial areas. Lemma: If a straight line is cut into unequal parts, then the sum of the squareson the unequal parts is greater than twice the rectangle contained by theunequal parts.
60. The square on the binomial straight line applied to a rational straight line produces as breadth the first binomial.
61. The square on the first bimedial straight line applied to a rational straight line produces as breadth the second binomial.
62. The square on the second bimedial straight line applied to a rational straight line produces as breadth the third binomial.
63. The square on the major straight line applied to a rational straight line produces as breadth the fourth binomial.
64. The square on the side of a rational plus a medial area applied to a rational straight line produces as breadth the fifth binomial.
65. The square on the side of the sum of two medial areas applied to a rational straight line produces as breadth the sixth binomial.
66. A straight line commensurable with a binomial straight line is itself also binomial and the same in order.
67. A straight line commensurable with a bimedial straight line is itself also bimedial and the same in order.
68. A straight line commensurable with a major straight line is itself also major.
69. A straight line commensurable with the side of a rational plus a medial area is itself also the side of a rational plus a medial area.
70. A straight line commensurable with the side of the sum of two medial areas is the side of the sum of two medial areas.
71. If a rational and a medial are added together, then four irrational straight lines arise, namely a binomial or a first bimedial or a major or a side of a rational plus a medial area.
72. If two medial areas incommensurable with one another are added together, then the remaining two irrational straight lines arise, namely either a second bimedial or a side of the sum of two medial areas.
73. If from a rational straight line there is subtracted a rational straight line commensurable with the whole in square only, then the remainder is irrational; let it be called an apotome.
74. If from a medial straight line there is subtracted a medial straight line which is commensurable with the whole in square only, and which contains with the whole a rational rectangle, then the remainder is irrational; let it be called first apotome of a medial straight line.
75. If from a medial straight line there is subtracted a medial straight line which is commensurable with the whole in square only, and which contains with the whole a medial rectangle, then the remainder is irrational; let it be called second apotome of a medial straight line.
76. If from a straight line there is subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them added together rational, but the rectangle contained by them medial, then the remainder is irrational; let it be called minor.
77. If from a straight line there is subtracted a straight line which is incommensurable in square with the whole, and which with the whole makes the sum of the squares on them medial but twice the rectangle contained by them rational, then the remainder is irrational; let it be called that which produces with a rational area a medial whole.
78. If from a straight line there is subtracted a straight line which is incommensurable in square with the whole and which with the whole makes the sum of the squares on them medial, twice the rectangle contained by them medial, and further the squares on them incommensurable with twice the rectangle contained by them, then the remainder is irrational; let it be called that which produces with a medial area a medial whole.
79. To an apotome only one rational straight line can be annexed which is commensurable with the whole in square only.
80. To a first apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a rational rectangle.
81. To a second apotome of a medial straight line only one medial straight line can be annexed which is commensurable with the whole in square only and which contains with the whole a medial rectangle.
82. To a minor straight line only one straight line can be annexed which is incommensurable in square with the whole and which makes, with the whole, the sum of squares on them rational but twice the rectangle contained by them medial.
83. To a straight line which produces with a rational area a medial whole only one straight line can be annexed which is incommensurable in square with the whole straight line and which with the whole straight line makes the sum of squares on them medial but twice the rectangle contained by them rational.
84. To a straight line which produces with a medial area a medial whole only one straight line can be annexed which is incommensurable in square with the whole straight line and which with the whole straight line makes the sum of squares on them medial and twice the rectangle contained by them both medial and also incommensurable with the sum of the squares on them.