Proceedings of the Edinburgh Mathematical Society, Volume 16Scottish Academic Press, 1898 - Electronic journals |
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Page 2
... difference is to be found in the almost exclusive use by Archimedes of inequality theorems which are required for the application of the method of exhaustion , while the modern treatment replaces this by a more or less rigorous use of ...
... difference is to be found in the almost exclusive use by Archimedes of inequality theorems which are required for the application of the method of exhaustion , while the modern treatment replaces this by a more or less rigorous use of ...
Page 3
... difference , and the number of terms , as the real simplicity of the proofs is thereby obscured . The chief defect of his notation - and it is one that causes great prolixity both in statement and in demonstration -- is the absence of a ...
... difference , and the number of terms , as the real simplicity of the proofs is thereby obscured . The chief defect of his notation - and it is one that causes great prolixity both in statement and in demonstration -- is the absence of a ...
Page 4
... difference ; it will be seen later how Archimedes gets over that restriction when a series occurs not satisfying that condition . 4. The most important theorems are those dealing with the sums of squares , and it was possibly the ...
... difference ; it will be seen later how Archimedes gets over that restriction when a series occurs not satisfying that condition . 4. The most important theorems are those dealing with the sums of squares , and it was possibly the ...
Page 6
... difference : — 3 ( a2 + b2 + ... + 12 ) = ( n + 1 ) a2 + l ( a + b + ... + 1 ) and the corollary is 3 ( a2 + b2 + ... + 12 ) > na2 > 3 ( b2 + ... + 12 ) The proof is very peculiar . It is obvious that a = b + l = c + k = d + j = etc ...
... difference : — 3 ( a2 + b2 + ... + 12 ) = ( n + 1 ) a2 + l ( a + b + ... + 1 ) and the corollary is 3 ( a2 + b2 + ... + 12 ) > na2 > 3 ( b2 + ... + 12 ) The proof is very peculiar . It is obvious that a = b + l = c + k = d + j = etc ...
Page 7
... difference to be equal to the least term , but obviously cases arise where that condition is not satisfied , and Archimedes provides for such cases in the 11th proposition of the same book ( Opera II . , 42-50 ) . The diagram to that ...
... difference to be equal to the least term , but obviously cases arise where that condition is not satisfied , and Archimedes provides for such cases in the 11th proposition of the same book ( Opera II . , 42-50 ) . The diagram to that ...
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Common terms and phrases
abscisses algebraic algorithmes naturels angles Archimedes axes axis B.Sc b₁ Betti's Theorem centre College common difference Conformal Transformation conic constants corresponding COSC courbe D.Sc d'une d²r denote deux racines displacements dy dz Edinburgh égale equal equation équations Euclid's definition exemple fonction force parallel geometry George Heriot's School George Watson's College given Glasgow Hence hyperlogarithmes inequalities integral irrational number l'algorithme l'autre l'équation l'expression l'on LAWRENCE CRAWFORD LL.D logarithme magnitudes Mathematics membre method moyen des algorithmes nombre ordonnée peut plane pourra premier Professor proportion proposition puissance qu'on quadratic space quantuplicity racines réelles ratio rational numbers relation représenter RITCHIE SCOTT School segment singular solutions solid Stewart's theorem suppose surface tractions surpuissance symbole théorème theory triangle U₁ unaltered unit force v₁ valeur w₁
Popular passages
Page 62 - ... EDINBURGH. Mathematical Society, May 13. — Mr. JB Clark, President in the chair. — The following papers were read :— On the second solutions of Lame's equation, by Mr. Lawrence Crawford (communicated by Mr. JW Butters); on the insolation of a sun of sensible magnitude, by Mr. A. Ritchie Scott; the singular solutions of a certain differential equation of the second order, by Mr. Hugh Mitchell. PARIS. Academy of Sciences, June 6. — M. Wolf in the chair. — New photographic studies of the...
Page 37 - In spherical triangles, whether right angled or oblique angled, the sines of the sides are proportional to the sines of the angles opposite to them.
Page 98 - II. A greater magnitude is said to be a multiple of a less, when the greater is measured by the less, that is, ' when the greater contains the less a certain number of times exactly.' III. " Ratio is a mutual relation of two magnitudes of the same kind to one another, in respect of quantity.
Page 45 - Stewart's theorem enables us to resolve easily the following problem : " To draw a circle touching another given circle and passing through two given points.
Page 79 - ... produced. The proposition, however, is equally true for both figures. By adopting the suggested convention we get rid both of the difficulty of drawing the figure and of the implication that the truth of the proposition is limited to the case where the perpendiculars lie on opposite sides of BC. (c) " AB, BC are equal arcs of a circle and P is a point on the arc BC, show that BP bisects the angle contained by AP and CP produced.
Page 51 - ... methods conducted to the same result. Object of the present paper is the analytical demonstration and the extension to «-dimensional space of a third mode of generation, equally due to Mannheim, according to which the wave-surface is the locus of the point that admits with respect to a given ellipsoid an enveloping cone , one of the principal sections of which is a right angle. This generalisation however does not conduct to the same « — 1-dimensional figure as before (p.