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adjacent angle algebra analytical units apply arithmetic assertion AUGUSTUS DE MORGAN 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 right angle 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 61 - C, which is compounded of the ratios of A to B, and B to C, is the same with the ratio of D to F, which is compounded of the ratios of D to E, and E to F.
Page 89 - PRINCIPLES OF GEOMETRY, familiarly Illustrated, and applied to a variety of useful purposes. Designed for the Instruction of Young Persons.
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 89 - THE STUDENT'S JOURNAL. Arranged, Printed, and Ruled for receiving an Account of every Day's Employment for the space of One Year. With an Index and Appendix.
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 89 - NATURAL PHILOSOPHY FOR BEGINNERS. Being familiar Illustrations of the Laws of Motion and Mechanics, intended as a Text Book for Schools and Self-instruction, as a Companion to the Lecture Room, or for Model Schools. Illustrated with 143 Engravings on Wood.
Page 147 - The Connexion of Number and Magnitude; An attempt to explain the fifth book of Euclid.
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 73 - ... proficient in a symbolic calculus would naturally demand a supply of meaning."1 And again: "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 could give any trouble; .... The next and second step, .... consisted in treating the results of algebra...