...as the " Common Notions," so far as they relate to quantity, prefixed by Euclid to his Elements. " Things which are equal to the same are equal to one another." " If equals be added to equals, the wholes are equal." " If equals be taken from equals, the remainders...

...in the first place, what are called the Axioms of Mathematics or the Science of Quantity, such as " Things which are equal to the same, are equal to one another." " If equals be added to equals, or subtracted from equals, the wholes, or the remainders, will be equal."...

...dilutions, and the pure ideal. The result would determine whether the dictum of opponents be just, that things which are equal to the same are equal to one another. So much by way of exordium. I now proceed to the subject in hand. The propriety of alternating medicines...

...dilutions, and the pure ideal. The result would determine whether the dictum of opponents be just, that things which are equal to the same are equal to one another. So much by way of exordium. I now proceed to the subject in hand. The propriety of alternating medicines...

...insisted on by Mr. Lewes and others : namely, that alcohol replaced a certain amount of food ; and " as things which are equal to the same are equal to one another," he inferred that if a glass of ale was equal to a slice of mutton, in its satisfying effect, and that...

...etymologically, and with the latter, territorially ; and not forgetting the timehonoured axiom of Euclid, that " things which are equal to the same are equal to one another :" the conclusion seems irresistible, that lim and Azem are but component parts of a single proper...

...assailants should find such difficulty in grasping so palpahle a truism as the first axiom of Euclid, that " things which are equal to the same are equal to one another," or should refuse to apply it to lines and curves and geometrical figures. They even reverse it when...

...Euclid are felt to be true in every age, and among every people : "The whole is greater than its part." "Things which are equal to the same, are equal to one another." There may be truths which reason can neither discover nor comprehend; but nothing can be true which...

...mentions that Apollonius attempted to prove the axioms, and cites his investigation of the theorem, that things which are equal to the same are equal to one another, in which, as may be supposed, propositions are assumed not more obvious than the theorem itself. Vitruvius...

...principle in all such cases is either, ' Things are the same which are the same with a third,' or ' things which are equal to the same are equal to one another.' Much confusion is avoided by allotting reasoning of this description to a separate head. As there is...