...the reasoning races of men. For instance, two of these are ' The whole is greater than its part ; ' ' If things are equal to the same thing, they are equal to one another.' He would, it may be presumed, argue — ' " Whole," " part," and " greater," being understood,...

...and the only mark which designates the space between them is the algebraic sign of equality. Being equal to the same thing, they are equal to each other; and the man who stands aloof and imagines by his excellence that he is a law unto himself, that he serves...

...number, the products are equal. 8. If equals are divided by the same number, the quotients are equal. 9. If things are equal to the same thing, they are equal to each other. 10. The same parts of equals are equal. 11. Equal powers of equals are equal. 15. Notation (Latin notare,...

...start from undemonstrable principles. Otherwise the steps of a demonstration would be endless." Euclid divided unproved propositions into two classes : axioms,...sentences, usually giving a separate line to each statemerit, followed by the definition, axiom, or theorem on which it depends. For this purpose the...

...longer necessary to hold that there must be more than a verbal difference behind the forms ' if two things are equal to the same thing they are equal to each other ' and ' all things equal to the same thing are equal to each other.' Neither, on the other hand, shall we...

...other because they are each equal to x. This principle may be stated as follows: If two quantities are equal to the same thing, they are equal to each other. 177 This method of solving simultaneous equations is called Comparison. fx3 — v3=127 m Examples:...

...same manner, it can be shown that , , = TDTV Hince the two left-hand members of these two equations are equal to the same thing, they are equal to each other, and 0a Oa' OA" Ob Ob' OB" In other words, when the lines of action of the power and weight are parallel,...

...Experience is further defined as " the process of becoming expert by experiment.'' Since these terms are equal to the same thing they are equal to each other. But how can a process be a " substance " (mind) or possess a " faculty " ? The major premiss of the...

...now than previously, or the converse. Similarly, this is true: "whenever there are some things which are equal to the same thing, they are equal to each other." This, however, is false: "Whenever there are such that each of them was equal to the same quantity,...

...expression of how its empty abstractness functions can be found in the mathematical formula, "If two things are equal to the same thing, they are equal to each other," for here the three terms are abstracted from all qualitative characteristics and considered only in...