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adjacent angle algebra analytical units apply arithmetic assertion AUGUSTUS DE MORGAN binomial theorem called circle coincide commensurable Consequently consider contained cos0 cosC cosecant cosine cosx 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 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 89 - Axis, of a sphere is a line passing through the centre, 'and terminated both ways by the surface, as the line D E.
Page 33 - 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 77 - ... phrase. Nothing could make a more easy pillow for the mind, than the rejection of all which could give any trouble ; . . . . 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...
Page 26 - 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 148 - The Connexion of Number and Magnitude; An attempt to explain the fifth book of Euclid.
Page 58 - ... 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 76 - Vitium negationis, was his phrase. Nothing could make a more easy pillow for the mind, than the rejection of all which could give any trouble; .... 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...
Page 77 - This ought to have been the most startling part of the whole process. That contradictions might occur, was no wonder ; but that contradictions should uniformly, and without exception, lead to truth in algebra, and in no other species of mental occupation whatsoever, was a circumstance worthy the name of a mystery. Nothing could prevail against the practical result that theorems so produced were true ; and at last, when the interpretation of the abstract negative quantity shewed that a part at least...