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abelian groups algebraic angle arithmetic arrows axiomatic axioms bijection calculus called cardinal number Chapter charts circle complex numbers composite congruence construction continuous convergent coordinates corresponding cotangent curvature curve deﬁned deﬁnition described differential diﬂerent diﬂerential dimensional elements equations Euclidean geometry example ﬁeld Figure ﬁgures ﬁnd ﬁnite ﬁrst ﬁxed formal deﬁnition formula function f functor Galois given Hence holomorphic function idea inﬁnite inﬁnitesimal inner product integral interval inverse isomorphism magnitudes manifold Mathematics matrix means metric space modulo multiplication natural numbers number theory objects open set operations ordered set ordinal path Peano Peano postulates permutation plane polynomial positive power series proof properties quantiﬁers rational numbers real numbers reﬂection Riemann surface rigid motion rotation rules satisﬁes scalar segment sequence set theory smooth functions Speciﬁcally sphere structure subgroup subset symmetry tangent theorem tion topological space transformation group triangle unique values variable vector space velocity zero