These things being assumed, it is proved that there exist an infinite multitude of straight lines commensurable and incommensurable with an assigned straight line - those (commensurable and incommensurable) in length only, and those also (commensurable or incommensurable) in square)^{1}. Therefore, let the assigned straight line be called rational. And (let) the (straight lines) commensurable with it, either in length and square, or in square only, (also be called) rational. But let the (straight lines) incommensurable with it be called irrational.^{2}
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Corollaries: 1 2
Definitions: 3 4 5 6 7 8 9 10
Lemmas: 11
Proofs: 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110
Propositions: 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192
To be more exact, straight lines can either be commensurable in square only, incommensurable in length only, or commensurable/incommensurable in both length and square, with an assigned straight line (translator's note). ↩
Let the length of the assigned straight line be unity. Then rational straight lines have lengths expressible as \(k\) or \(\sqrt{k}\), depending on whether the lengths are commensurable in length, or in square only, respectively, with unity. All other straight lines are irrational (translator's note). ↩