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adjacent angle algebra analytical units apply arithmetic assertion AUGUSTUS DE MORGAN binomial theorem called circle commensurable Consequently consider contained cos0 cos2 cosC cosecant cosine cotangent deduced definition denoted derived function diminishes without limit distinction equal equation exceed expressed Fifth Book follows formula geometry given gives greater ratio Hence hypothenuse idea incommensurable increase infinite number instance inverse functions length less ratio lies linear unit logarithm magnitude mean proportionals method metic multiple scale namely nearly notion number or fraction operations polygon positive preceding primary functions proceed proposition proved question radius result roots of unity secant shew shewn side Similarly sin0 sin2 sine square root straight line student subdivision subtraction suppose supposition symbol tangent theorem thing third triangle Trigonometry true unity whence whole number
Page 23 - B'. The failure of the second law of addition is due (as we should say now that we have investigated the matter) to the fact that the bodies themselves have a finite heat capacity and do not give up all their heat to the water; the second law will not be true unless the sum of the heat capacities of A and B is the same as that of A
Page 53 - That magnitude which has a greater ratio than another has unto the same magnitude, is the greater of the two : and that magnitude to which the same has a greater ratio than it has unto another magnitude, is the lesser of the two.
Page 30 - 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 second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.
Page 145 - The Connexion of Number and Magnitude; An attempt to explain the fifth book of Euclid.
Page 73 - Ignorance, the necessary predecessor of knowledge, was called nature; and all conceptions which were declared unintelligible by the former, were supposed to have been made impossible by the latter. The first who used algebraical symbols in a general sense, Vieta, concluded that subtraction was a defect, and that expressions containing it should be in every possible manner avoided. Vitium negationis, was his phrase. Nothing could make a more easy pillow for the mind, than the rejection of all which...
Page 55 - ... F is greater than E, but not greater than D (V. Def. 7). Because E and D are equimultiples of B and A, and E is less than D . (Const). Therefore B is less than A (V Ax. 4) Therefore, of two magnitudes, &o QED PBOP. XI. THEOREM. Ratios that are equal to the same ratio, are equal to one another. If A is to B as C is to D ; and C is to D, as E is to F.
Page 74 - The next and second step, .... consisted in treating the results of algebra as necessarily true, and as representing some relation or other, however inconsistent they might be with the suppositions from which they were deduced. So soon as it was shewn that a particular result had no existence as a quantity, it was permitted, by definition, to have an existence of another kind, into which no particular inquiry was made, because the rules under which it was found that the new symbols would give true...
Page 58 - If A : B = C : D, C and D being less than A and B, then A- C : B— D = A : B.