The Elements of Euclid |
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Page 120
... proportionals , it is usually expressed by saying , the first is to the se- ' cond , as the third to the fourth ... proportionals , the first is said to have to the third the duplicate ratio of that which it has to the second . XI . When ...
... proportionals , it is usually expressed by saying , the first is to the se- ' cond , as the third to the fourth ... proportionals , the first is said to have to the third the duplicate ratio of that which it has to the second . XI . When ...
Page 121
... proportionals , so as that they continue still to be proportionals . ' XIII . Permutando , or alternando , by permutation , or alternately ; this See Note . word is used when there are four proportionals , and it is inferred , that the ...
... proportionals , so as that they continue still to be proportionals . ' XIII . Permutando , or alternando , by permutation , or alternately ; this See Note . word is used when there are four proportionals , and it is inferred , that the ...
Page 122
... proportionals when taken two and two of each rank , and it is inferred , that the first is to the last of the first rank of magnitudes , as the first is to the last of the others : Of this there are the two following kinds , which arise ...
... proportionals when taken two and two of each rank , and it is inferred , that the first is to the last of the first rank of magnitudes , as the first is to the last of the others : Of this there are the two following kinds , which arise ...
Page 129
... proportionals , they are pro- See Note . portionals also when taken inversely . If the magnitude A be to B , as C is to D , then also inversely B is to A , as D to C. Take of B and D any equimultiples whatever E and F ; and of A and C ...
... proportionals , they are pro- See Note . portionals also when taken inversely . If the magnitude A be to B , as C is to D , then also inversely B is to A , as D to C. Take of B and D any equimultiples whatever E and F ; and of A and C ...
Page 136
... proportionals , as one of the antecedents is to its consequent , so shall all the antecedents taken together be to all the conse- quents . Let any number of magnitudes A , B , C , D , E , F be propor- tionals ; that is , as A is to B ...
... proportionals , as one of the antecedents is to its consequent , so shall all the antecedents taken together be to all the conse- quents . Let any number of magnitudes A , B , C , D , E , F be propor- tionals ; that is , as A is to B ...
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Common terms and phrases
altitude angle ABC angle BAC base BC BC is equal BC is given bisected Book XI centre circle ABCD circumference cone cylinder demonstrated described diameter draw drawn equal angles equiangular equimultiples Euclid excess fore given angle given in magnitude given in position given in species given magnitude given ratio given straight line gnomon greater join less Let ABC meet multiple parallel parallelogram parallelogram AC perpendicular point F polygon prism proportionals proposition pyramid Q. E. D. PROP radius ratio of AE rectangle CB rectangle contained rectilineal figure remaining angle right angles segment sides BA similar solid angle solid parallelopiped square of AC straight line AB straight line BC tangent THEOR third triangle ABC triplicate ratio vertex wherefore