The First Six Books of the Elements of Euclid: With a Commentary and Geometrical Exercises. To which are Annexed a Treatise on Solid Geometry, and Short Essays on Ancient Geometrical Analysis, and the Theory of Transversals |
Common terms and phrases
A B and B C altitude angle A B C axiom bisect Book centre circumference circumscribed coincide cone conical surface const constructed cylinder demonstration described diagonal diameter difference draw equal angles equal to twice equiangular equilateral triangle equimultiples Euclid external angle extremities geometry given circle given line given point given right line greater Hence homologous sides hypotenuse inscribed intersect isosceles triangle less magnitude manner mean proportional multiple opposite parallel parallelogram parallelopiped pentagon perpendicular plane polygon prism problem produced PROPOSITION proved pyramid quadrilateral radii radius ratio rectangle rectilinear figure respectively equal right angles right line A B segment sides A B similar solid angle square of A B surface tangent THEOREM third tiples triangle A B C vertex vertical angle
Popular passages
Page 22 - If two triangles have two sides of the one respectively equal to two sides of the other, and the contained angles supplemental, the two triangles are equal.
Page 30 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Page 38 - DE : but equal triangles on the same base and on the same side of it, are between the same parallels ; (i.
Page 152 - ... If there be three magnitudes, and other three, which have the same ratio taken two and two, but in a cross order; then if the first magnitude be greater than the third, the fourth shall be greater than the sixth: and if equal, equal; and if less, less.
Page 141 - Let it be multiplied until it become greater than D, and let the other be multiplied as often ; and let EF be the multiple thus taken of AC, and FG the same multiple of CB : therefore...
Page 135 - Let A the first have to B the second the same ratio which the third C has to the fourth D ; and of A and C | let...
Page 144 - N ; and if equal, equal ; and if less, less : but if G be greater than L, it has been shown that L HC K E M F N H is greater than M ; and if equal, equal; and if less, less: therefore, if G be greater than L, K is greater than N ; and if equal, equal ; and if less less : and G, K are any equimultiples whatever of A, E ; and L, N any whatever of B, F : therefore as A is to B, so is E to F (5.
Page 306 - If two angles of one triangle are equal to two angles of another triangle, the third angles are equal, and the triangles are mutually equiangular.
Page 139 - IF the first be to the second as the third to the fourth, and if the first be a multiple, or part of the second; the third is the same multiple, or the same part of the fourth...
Page 127 - And the same thing is to be understood when it is more briefly expressed by saying, a has to d the ratio compounded of the ratios of e to f, g to h, and k to l. In like manner, the same things being supposed, if m has to n the same ratio which a has to d ', then, for shortness...