...XXаiгXаffккi/юг, írartpov íKa.Ttpov if Ира uiгípíxy, í) «/ia tffa y, s"/ ¿'/ia tXXeíirç KaraXXr;Xa. Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth ; when any equimultiples whatsoever of the first and third, compared with...

...; and with the view of removing this objection, Elrington has substituted the following, namely, " Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth, when any submultiple whatsoever of the first is contained in the second,...

...greater than that of the second, the multiple of the third is also greater than that of the fourth. " Magnitudes are said to be in the same ratio, the first...the second, and the third to the fourth, — when the equimultiples of the first and third, being at the same time compared with the equimultiples of...

...iro\\air\aaiaafiov, екarfpov fKuTBpov jj аfia inrepk%y, jj afia laa y, i) ¡ífia éXXeiirp KaráXXqXa. Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth; when any equimultiples whatsoever of the first and third, compared with...

...FOVttTH BOOK. FIFTH BOOK. DEFINITIONS. 1. A less magnitude is called an aliquot part, or a lubmultiple of a greater, when the less measures the greater....the first is contained in the second, as often as an equi-submultiple of the third is contained in the fourth. 6. Magnitudes which have the same ratio are...

...substituted the following, namely, " Magnitudes are said to be in the same ratio, the first to the second as the third to the fourth, when any submultiple whatsoever...the first is contained in the second, as often as an equi-submultiple of the third is contained in the fourth." On the other hand, many of the most able...

...quantities given by Kuclid is as follows: — 'Magnitudes are said to have the same ratio to one another, the first to the second, and the third to the fourth, when equimultiples of the first and third, and equimultiples of the second and fourth, whatever the multiplications...

...very theorems it is desired to prove. Euclid,s definition of proportion is as follows : Quantities are said to be in the same ratio, the first to the second and the third to the fourth when equal multiples of the first and third are at the same time greater equal or less than equal multiples...

...are said to have a ratio to one another which are capable, when multiplied, of exceeding one another. 5. Magnitudes are said to be in the same ratio, the...first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the...

...are said to have a ratio to one another which are capable, when multiplied, of exceeding one another. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the...