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, III

Definitions

1. Given a rational straight line and an apotome, if the square on the whole is greater than the square on the annex by the square on a straight line commensurable in length with the whole, and the whole is commensurable in length with the rational line set out, let the apotome be called a first apotome.
2. But if the annex is commensurable with the rational straight line set out, and the square on the whole is greater than that on the annex by the square on a straight line commensurable with the whole, let the apotome be called a second apotome.
3. But if neither is commensurable in length with the rational straight line set out, and the square on the whole is greater than the square on the annex by the square on a straight line commensurable with the whole, let the apotome be called a third apotome.
4. Again, if the square on the whole is greater than the square on the annex by the square on a straight line incommensurable with the whole, then, if the whole is commensurable in length with the rational straight line set out, let the apotome be called a fourth apotome;
5. If the annex be so commensurable, a fifth;
6. And, if neither, a sixth.

Propositions

85. To find the first apotome.
86. To find the second apotome.
87. To find the third apotome.
88. To find the fourth apotome.
89. To find the fifth apotome.
90. To find the sixth apotome.
91. If an area is contained by a rational straight line and a first apotome, then the side of the area is an apotome.
92. If an area is contained by a rational straight line and a second apotome, then the side of the area is a first apotome of a medial straight line.
93. If an area is contained by a rational straight line and a third apotome, then the side of the area is a second apotome of a medial straight line.
94. If an area is contained by a rational straight line and a fourth apotome, then the side of the area is minor.
95. If an area is contained by a rational straight line and a fifth apotome, then the side of the area is a straight line which produces with a rational area a medial whole.
96. If an area is contained by a rational straight line and a sixth apotome, then the side of the area is a straight line which produces with a medial area a medial whole.
97. The square on an apotome of a medial straight line applied to a rational straight line produces as breadth a first apotome.
98. The square on a first apotome of a medial straight line applied to a rational straight line produces as breadth a second apotome.
99. The square on a second apotome of a medial straight line applied to a rational straight line produces as breadth a third apotome.
100. The square on a minor straight line applied to a rational straight line produces as breadth a fourth apotome.
101. The square on the straight line which produces with a rational area a medial whole, if applied to a rational straight line, produces as breadth a fifth apotome.
102. The square on the straight line which produces with a medial area a medial whole, if applied to a rational straight line, produces as breadth a sixth apotome.
103. A straight line commensurable in length with an apotome is an apotome and the same in order.
104. A straight line commensurable with an apotome of a medial straight line is an apotome of a medial straight line and the same in order.
105. A straight line commensurable with a minor straight line is minor.
106. A straight line commensurable with that which produces with a rational area a medial whole is a straight line which produces with a rational area a medial whole.
107. A straight line commensurable with that which produces a medial area and a medial whole is itself also a straight line which produces with a medial area a medial whole.
108. If from a rational area a medial area is subtracted, the side of the remaining area becomes one of two irrational straight lines, either an apotome or a minor straight line.
109. If from a medial area a rational area is subtracted, then there arise two other irrational straight lines, either a first apotome of a medial straight line or a straight line which produces with a rational area a medial whole.
110. If from a medial area there is subtracted a medial area incommensurable with the whole, then the two remaining irrational straight lines arise, either a second apotome of a medial straight line or a straight line which produce with a medial area a medial whole.
111. The apotome is not the same with the binomial straight line.
112. The square on a rational straight line applied to the binomial straight line produces as breadth an apotome the terms of which are commensurable with the terms of the binomial straight line and moreover in the same ratio; and further the apotome so arising has the same order as the binomial straight line.
113. The square on a rational straight line, if applied to an apotome, produces as breadth the binomial straight line the terms of which are commensurable with the terms of the apotome and in the same ratio; and further the binomial so arising has the same order as the apotome.
114. If an area is contained by an apotome and the binomial straight line the terms of which are commensurable with the terms of the apotome and in the same ratio, then the side of the area is rational.
     Corollary: It is possible for a rational area to be contained by irrationalstraight lines.
115. From a medial straight line there arise irrational straight lines infinite in number, and none of them is the same as any preceding.